by Hoi-Kwan Lau - University of Toronto T-Space...Prof. Hoi-Kwong Lo. I really appreciate that they agreed to co-supervise my PhD thesis; this was an adventurous decision as they work

Practicality of Quantum Information Processing

by

Hoi-Kwan Lau

A thesis submitted in conformity with the requirementsfor the degree of Doctor of Philosophy

Graduate Department of PhysicsUniversity of Toronto

c© Copyright 2014 by Hoi-Kwan Lau

Abstract

Practicality of Quantum Information Processing

Hoi-Kwan Lau

Doctor of Philosophy

Graduate Department of Physics

University of Toronto

2014

Quantum Information Processing (QIP) is expected to bring revolutionary enhancement

to various technological areas. However, today’s QIP applications are far from being

practical. The problem involves both hardware issues, i.e., quantum devices are imper-

fect, and software issues, i.e., the functionality of some QIP applications is not fully

understood.

Aiming to improve the practicality of QIP, in my PhD research I have studied various

topics in quantum cryptography and ion trap quantum computation. In quantum cryp-

tography, I first studied the security of position-based quantum cryptography (PBQC).

I discovered a wrong assumption in the previous literature that the cheaters are not al-

lowed to share entangled resources. I proposed entanglement attacks that could cheat all

known PBQC protocols.

I also studied the practicality of continuous-variable (CV) quantum secret sharing

(QSS). While the security of CV QSS was considered by the literature only in the limit

of infinite squeezing, I found that finitely squeezed CV resources could also provide finite

secret sharing rate. Our work relaxes the stringent resources requirement of implementing

QSS.

In ion trap quantum computation, I studied the phase error of quantum information

induced by dc Stark effect during ion transportation. I found an optimized ion trajectory

for which the phase error is the minimum. I also defined a threshold speed, above which

ion transportation would induce significant error.

ii

In addition, I proposed a new application for ion trap systems as universal bosonic

simulators (UBS). I introduced two architectures, and discussed their respective strength

and weakness. I illustrated the implementations of bosonic state initialization, transfor-

mation, and measurement by applying radiation fields or by varying the trap potential.

When comparing with conducting optical experiments, the ion trap UBS is advantageous

in higher state initialization efficiency and higher measurement accuracy.

Finally, I proposed a new method to re-cool ion qubits during quantum computation.

The idea is to transfer the motional excitation of a qubit to another ion that is prepared

in the motional ground state. I showed that my method could be ten times faster than

current laser cooling techniques, and thus could improve the speed of ion trap quantum

computation.

iii

Acknowledgements

First of all, may I express my sincere gratitude to my supervisors: Prof. Daniel James and

Prof. Hoi-Kwong Lo. I really appreciate that they agreed to co-supervise my PhD thesis;

this was an adventurous decision as they work in different directions. I am very fortunate

to have had numerous inspiring discussions with my supervisors, through which I have

learnt a lot in both physics knowledge and the ways to tackle problems. I am obliged for

their patience and tolerance, particularly they relentlessly gave me friendly advices even

when I held a strong opposing opinion. Besides, I would like to thank for their effort and

consideration in helping me in job-hunting and in training me to be a better physicist

and person. In summary, my PhD period would not be so fruitful and enjoyable without

the kindness of my supervisors.

I would like to thank my supervisory committee members, Joseph Thywissen and John

Sipe, and my examiners, Jonathan Dowling and John Wei, for their valuable comments

and beneficial advices. I also want to thank my colleagues, including Eric Chitambar,

Serge Fehr, John Gaebler, Barry Sanders, Marcos Villagra, Christian Weedbrook, and

Andrew White, for their illuminating discussions and useful helps. In addition, I want

to thank Mr. David Tang and Ms. Kristy Kwok for their hospitality during my stay in

Toronto.

It is my honour to acknowledge the support from the Kwok Sau Po Scholarship, the

E. F. Burton Fellowship, the Lachlan Gilchrist Fellowship Fund, and the Queen Elizabeth

II Graduate Scholarship in Science and Technology.

Finally, I would like to thank my beloved wife, Dan Sun, for agreeing to accompanying

me in the rest of my life.

iv

Contents

Glossaries and Acronyms xi

1 Introduction 1

2 Position-Based Quantum Cryptography 6

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.2 Classical Position-Based Cryptrography . . . . . . . . . . . . . . . . . . 8

2.2.1 Individually Cheating . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.2.2 Collaborative Cheating . . . . . . . . . . . . . . . . . . . . . . . 10

2.3 PBQC Protocols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.3.1 Protocol A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.3.2 Protocol B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.4 Cheating in N = 2 Case . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.4.1 Cheating against Protocol A . . . . . . . . . . . . . . . . . . . . 12

2.4.2 Cheating against Protocol B . . . . . . . . . . . . . . . . . . . . . 15

2.5 Cheating in N > 2 Case . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.5.1 Cheating against Protocol A . . . . . . . . . . . . . . . . . . . . . 17

2.5.2 Cheating against Protocol B . . . . . . . . . . . . . . . . . . . . . 20

2.6 Principle of the cheating schemes . . . . . . . . . . . . . . . . . . . . . . 22

2.6.1 Protocol A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.6.2 Protocol B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.7 Modified PBQC Protocol . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3 Quantum Secret Sharing with Continuous-Variable Cluster States 28

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

3.2.1 Quantum Secret Sharing . . . . . . . . . . . . . . . . . . . . . . 30

v

3.2.2 Continuous-Variable Cluster States . . . . . . . . . . . . . . . . . 32

3.2.2.1 Nullifier representation . . . . . . . . . . . . . . . . . . 32

3.2.2.2 Wigner function representation . . . . . . . . . . . . . . 33

3.2.2.3 Correlations of measurement . . . . . . . . . . . . . . . 34

3.2.2.4 Cluster-class state . . . . . . . . . . . . . . . . . . . . . 35

3.3 CC Quantum secret sharing . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.4 CQ Quantum secret sharing . . . . . . . . . . . . . . . . . . . . . . . . . 38

3.4.1 Equivalence of CQ Quantum Secret Sharing and QKD . . . . . . 40

3.4.2 Secret Sharing Rate . . . . . . . . . . . . . . . . . . . . . . . . . 41

3.4.3 Simplification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

3.4.3.1 Mixed State Approach . . . . . . . . . . . . . . . . . . . 45

3.4.3.2 Classical Memory . . . . . . . . . . . . . . . . . . . . . . 46

3.4.3.3 Local Measurement . . . . . . . . . . . . . . . . . . . . . 48

3.4.3.4 Simplified CQ Protocol . . . . . . . . . . . . . . . . . . 50

3.5 QQ Quantum secret sharing . . . . . . . . . . . . . . . . . . . . . . . . . 51

3.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

4 Motional States of Trapped Ions 56

4.1 Trap and Ion Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

4.2 Internal Structure and Laser Operation . . . . . . . . . . . . . . . . . . 59

4.2.1 Transition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

4.2.2 Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

4.3 Ion Motion in Trap Potential . . . . . . . . . . . . . . . . . . . . . . . . 61

4.3.1 Generalized Harmonic Oscillator . . . . . . . . . . . . . . . . . . 63

4.3.2 Interaction Picture . . . . . . . . . . . . . . . . . . . . . . . . . . 67

5 Decoherence Induced by dc Electric field During Ion Transport 69

5.1 Speed of ion trap quantum computer . . . . . . . . . . . . . . . . . . . . 69

5.2 Motion of ion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

5.3 Phase shift due to dc Stark effect . . . . . . . . . . . . . . . . . . . . . . 72

5.4 Minimum possible phase shift . . . . . . . . . . . . . . . . . . . . . . . . 73

5.5 Threshold speed of transporting ion qubits . . . . . . . . . . . . . . . . . 76

5.6 Non-encoding State Excitation . . . . . . . . . . . . . . . . . . . . . . . 77

5.7 Summary and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

6 Ion Trap Bosonic Simulator 1: Multiple Ions in Single Trap 82

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

vi

6.2 Layout of the system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

6.3 Universal Bosonic Simulation . . . . . . . . . . . . . . . . . . . . . . . . 85

6.4 Laser Implementation of Basic Operations . . . . . . . . . . . . . . . . . 87

6.4.1 Displacement Operator . . . . . . . . . . . . . . . . . . . . . . . . 89

6.4.2 Phase-Shift Operator . . . . . . . . . . . . . . . . . . . . . . . . . 90

6.4.3 Squeezing Operator . . . . . . . . . . . . . . . . . . . . . . . . . . 90

6.4.4 Nonlinear Operator . . . . . . . . . . . . . . . . . . . . . . . . . . 90

6.4.5 Beam Splitter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

6.5 Readout . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

6.5.1 Adiabatic passage . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

6.5.2 Resonant Pulses . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

6.5.2.1 Post-selection Method . . . . . . . . . . . . . . . . . . . 100

6.5.2.2 Multiple Electronic State Method . . . . . . . . . . . . 103

6.6 Initialization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

7 Ion Trap Bosonic Simulator 2: Ions in Separate Trap 107

7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

7.2 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

7.3 Single Mode Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

7.3.1 Displacement Operator . . . . . . . . . . . . . . . . . . . . . . . 110

7.3.2 Squeezing Operator . . . . . . . . . . . . . . . . . . . . . . . . . 111

7.3.3 Phase-Shift Operator . . . . . . . . . . . . . . . . . . . . . . . . . 112

7.3.4 Nonlinear Operator . . . . . . . . . . . . . . . . . . . . . . . . . 113

7.4 Two-mode Operation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

7.4.1 Ion Transport and Pick-up . . . . . . . . . . . . . . . . . . . . . 119

7.4.2 Accuracy of beam splitter . . . . . . . . . . . . . . . . . . . . . . 120

7.5 Initialization and readout . . . . . . . . . . . . . . . . . . . . . . . . . . 122

7.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

8 Rapid ion re-cooling by swapping beam splitter 126

8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

8.2 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

8.3 Cooling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

8.4 Swapping Beam Splitter . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

8.4.1 Ansatz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

8.5 Ground state qubit pair . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

8.6 Transport between traps . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

vii

8.7 Implementation of potential . . . . . . . . . . . . . . . . . . . . . . . . . 137

8.8 Anharmonicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

8.9 Fluctuation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

8.10 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

9 Summary 142

9.1 Prospects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

A Appendix 148

A.1 Security of Modified Protocol . . . . . . . . . . . . . . . . . . . . . . . . 148

A.1.1 Security Against Attacks with One Entangled Qubit . . . . . . . 148

A.1.2 Security Against Attacks with One Entangled Qutrit . . . . . . . 151

A.2 Example of CC QSS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

A.2.1 Example 1: (2,3)-CC protocol . . . . . . . . . . . . . . . . . . . . 154

A.2.1.1 Parties 1,2 collaboration . . . . . . . . . . . . . . . . . 155

A.2.1.2 Parties 2,3 collaboration . . . . . . . . . . . . . . . . . 156

A.2.2 Example 2: (3,5)-CC protocol . . . . . . . . . . . . . . . . . . . . 157

A.2.2.1 Parties 1,2,3 collaboration . . . . . . . . . . . . . . . . 158

A.2.2.2 Parties 1,3,4 collaboration . . . . . . . . . . . . . . . . 159

A.3 Examples of CQ QSS . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

A.3.1 Example 1: (2,3)-CQ protocol . . . . . . . . . . . . . . . . . . . 161

A.3.1.1 Parties 1,2 collaboration . . . . . . . . . . . . . . . . 162

A.3.1.2 Parties 2,3 collaboration . . . . . . . . . . . . . . . . 164

A.3.2 Example 2: (3,5)-CQ protocol . . . . . . . . . . . . . . . . . . . 164

A.3.2.1 Parties 1,2,3 collaboration . . . . . . . . . . . . . . . 165

A.3.2.2 Parties 1,3,4 collaboration . . . . . . . . . . . . . . . 167

A.4 Example of QQ QSS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169

A.4.1 Example 1: (2,3)-QQ protocol . . . . . . . . . . . . . . . . . . . . 169

A.4.2 Example 2: (3,5)-QQ protocol . . . . . . . . . . . . . . . . . . . . 169

A.5 Example of Application: Demonstration of Hong-Ou-Mandel Effect . . . 171

A.6 Moving multiple ions by harmonic trap . . . . . . . . . . . . . . . . . . 174

A.6.1 Motional Excitation . . . . . . . . . . . . . . . . . . . . . . . . . 175

A.6.2 Example: Two ions in a trap . . . . . . . . . . . . . . . . . . . . . 178

Bibliography 181

viii

List of Figures

2.1 Space-time diagram of the one dimensional position-based cryptography

scenario. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.2 Illustration of the prover’s location . . . . . . . . . . . . . . . . . . . . . 10

2.3 Circuit for teleportation. . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.4 Scenario of three-verifier PBQC protocol . . . . . . . . . . . . . . . . . . 17

3.1 Strategy for computing the secret sharing rate using CV QKD techniques. 41

4.1 Layout of Paul trap. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

4.2 Energy levels of Raman transition . . . . . . . . . . . . . . . . . . . . . . 60

4.3 Energy levels under fluorescence measurement . . . . . . . . . . . . . . . 61

4.4 Layout of scalable ion trap quantum information processor . . . . . . . . 62

5.1 Time variation of optimal trajectories . . . . . . . . . . . . . . . . . . . . 75

6.1 Layout of bosonic simulator with multiple ions in a single trap . . . . . . 84

6.2 Schematic description of Raman transition. . . . . . . . . . . . . . . . . . 88

6.3 Minimum frequency difference between Raman field and unwanted mode

transition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

6.4 Probability fluctuation of phonon state under beam splitter operation with

different Lamb-Dicke parameter. . . . . . . . . . . . . . . . . . . . . . . . 93

6.5 Energy levels of ion during measurement process. . . . . . . . . . . . . . 96

6.6 List of elements in circuit diagram of boson simulation. . . . . . . . . . . 101

6.7 Circuit diagram of phonon non-resolving measurement using the post-

selection method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

6.8 Probability of the first 30 Fock states in the post-selected branch . . . . . 103

6.9 Circuit diagram of phonon number resolving measurement for nmax = 2 . 105

7.1 Configuration of ion trap UBS architecture involving ions in separated traps.109

7.2 Trap potential change for single mode operations . . . . . . . . . . . . . 110

ix

7.3 Procedure of a phonon beam splitter . . . . . . . . . . . . . . . . . . . . 115

7.4 Fidelity of phonon state after 50:50 beam splitter operation . . . . . . . . 121

8.1 Outline of the cooling process . . . . . . . . . . . . . . . . . . . . . . . . 128

8.2 Time variation of mean phonon number, ion separation, and local trap

parameters during SBS . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

8.3 Procedure of forming ground state qubit pair . . . . . . . . . . . . . . . . 135

8.4 Variations of potentials during diabatic ion separation . . . . . . . . . . . 136

8.5 Motional excitation of qubit induced by Coulomb anharmonicity . . . . . 139

8.6 Motional excitation caused by random fluctuation of potential . . . . . . 141

A.1 Schematic representation of the cluster states for the CC protocol. . . . . 155

A.2 Secret sharing rate of (2,3)- and (3,5)-CC protocols . . . . . . . . . . . . 160

A.3 Schematic representation of the cluster state for (2,3)- and the (3,5)-CQ

protocols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

A.4 Secret sharing rate of (2,3)- and (3,5)-CQ protocols . . . . . . . . . . . . 168

A.5 Entanglement extracted from CV cluster states in QQ QSS . . . . . . . . 170

A.6 Circuit diagram of the Hong-Ou-Mandel effect demonstration . . . . . . 172

x

Glossaries and Acronyms

access structure The set of authorised subsets of parties in a secret sharing protocol,

see Sec. 3.1 (p. 28).

adversary structure The set of unauthorised subsets of parties in a secret sharing

protocol, see Sec. 3.1 (p. 28).

CC Quantum secret sharing scheme that shares classical secret through secure quantum

channels, see Sec. 3.2.1 (p. 30).

cheater (PBQC) An unauthorised party who attempts to mimic the correct response

of the prover in PBQC, see Sec. 2.1 (p. 7).

CM Centre-of-mass phonon mode, see Sec. A.5 (p. 171).

CPHASE Controlled-phase gate, see Sec. 3.2.2 (p. 29).

CQ Quantum secret sharing scheme that shares classical secret through insecure quan-

tum channels, see Sec. 3.2.1 (p. 30).

CV Continuous-variable (p. 29).

dc Direct current (p. 69).

dealer The party who encodes a secret into some information carrier according to a

secret sharing protocol, see Sec. 3.1 (p. 28).

EPR Einstein-Podolsky-Rosen state (p. 38), see, e.g., Ref. [34].

GHZ Greenberger-Horne-Zeilinger state, see Eq. (2.2) in p. 12 and Ref. [15].

HOM Hong-Ou-Mandel effect (p. 104), see Ref. [86].

KMW Kielpinski-Monroe-Wineland architecture of ion trap quantum computer, see

Sec. 4.3 (p. 61).

LDA Lamb-Dicke approximation, see Sec. 6.4 (p. 89).

MBQC Measurement-Based Quantum Computation (p. 17), see Refs. [147, 148].

xi

nullifier See Sec. 3.2.2.1 (p. 32).

PBC Position-Based Cryptography, see Sec. 2.1 (p. 6).

PBQC Position-Based Quantum Cryptography, see Sec. 2.1 (p. 7).

prover The person whose position is to be verified in PBQC, see Sec. 2.1 (p. 6).

PVM Projection-valued measurement (p. 97).

QC Quantum computer (p. 56).

QIP Quantum Information Processing (p. 1).

QKD Quantum Key Distribution (p. 2).

QQ Quantum secret sharing scheme that shares quantum secret through insecure quan-

tum channels, see Sec. 3.2.1 (p. 30).

QSS Quantum Secret Sharing, see Sec. 3.1 (p. 28).

rf Radio frequency (p. 58).

RWA Rotating wave approximation, see Sec. 6.4 (p. 89).

SBS Swapping Beam Splitter, see Sec. 8.1 (p. 127).

state-averaging See p. 43.

UBS Universal Bosonic Simulator, see Sec. 6.1 (p. 83).

UQC Universal quantum computer (p. 82).

verifier Trusted reference station in PBQC, see Sec. 2.1 (p. 6).

xii

Chapter 1

Introduction

At the beginning of the 20th century, physicists formulated the theory of quantum me-

chanics. This new theory successfully explains numerous phenomena related to the low

energy “quantum” particles, such as atoms, electrons, and photons, whose properties

cannot be fully understood by using the classical theory of physics. However, quan-

tum mechanics also predicts that quantum particles would exhibit counter-intuitive be-

haviours. For example, unlike any everyday object whose physical attribute is definite, a

quantum particle may be present in a superposition state that simultaneously possesses

multiple different attributes. For instance, an electron in a superposition state can be

located inside two different quantum wells at the same time. Besides, unlike a classical

object that can be repeatedly measured without being disturbed, merely one shot of mea-

surement would erase the original state of a quantum particle. Furthermore, spatially

separated quantum particles can exhibit a correlation called entanglement that violates

some classical concepts of causality [57]. Such correlation cannot be explained by any

classical local hidden-variable theory [24, 25], thus entanglement is generally believed to

be a genuine physical effect.

The extraordinariness of quantum mechanics imposes difficulties for philosophers to

interpret the foundation of our world; on the other hand, scientists see the opportunities

and ask: Can we use these extraordinary quantum systems to build devices, which can

outperform current devices whose functionality is based on classical physics? The study

of Quantum Information Processing (QIP) aims to answer this question.

The principle of QIP is to use the states of quantum particles to encode information;

then the logical processing of information is conducted by applying physical operations

to transform the quantum states. In the past two decades, numerous QIP applications

have been discovered that are superior over their classical counterparts [143]. One QIP

application is to improve the security of communication. Because quantum information

1

Chapter 1. Introduction

must be disturbed by a measurement, any eavesdropping activities would leave a trace

that could be detected by the authorised parties. Using this property, a quantum key

distribution (QKD) protocol was proposed [26] and was proved to be secure against any

kind of eavesdropping [126, 120, 162], while such an unconditional security is impossible

for any classical communication scheme. Another application of QIP is to improve the

efficiency of computation. Since the size of the Hilbert space of a quantum system scales

exponentially with the number of particles involved, a quantum computer can process

an exponential amount of data in each operation. By using such capability, quantum

computational algorithms were invented and shown to be advantageous over any known

classical algorithm in numerous tasks, such as factoring larger numbers [161], searching

for a particular data from a library [73], and simulating complicated physical systems

[61, 116, 41].

Recently, quantum communication has been realised between stations that are over

100 km apart [171], and quantum computation involving 100 operations has been demon-

strated [104]. Nevertheless, current QIP applications are still far from being practically

useful. The problem involves both “hardware” and “software” issues. Firstly, it is difficult

to build a QIP device that functions as desired. Due to respective physical properties,

each quantum system suffers from limitations on the operations that can be efficiently

implemented. For example, due to the weak interaction between photons, entanglement

operations in optical quantum computers have to be implemented by non-deterministic

methods whose success rate is low. Besides, realistic quantum devices are imperfect and

are exposed to environmental influence; even a tiny flaw of control or a little background

noise would significantly affect the volatile quantum states, and thus contaminate the

encoded information. Furthermore, in spite of significant breakthroughs in recent years,

the speed of state-of-the-art quantum operations are generally much slower than that

of classical devices, whose technologies are already mature. All of these hardware is-

sues would hinder a realistic quantum device from attaining the theoretical efficiency of

quantum computational algorithms, and the ideal security of quantum communication

protocols.

Apart from the hardware issues, the practicality of some QIP applications is still

controversial. For example, it is not uncommon that seemingly promising QIP proposals

were later found to be not functioning as predicted. The problems usually originate from

making false assumptions due to classical perceptions. For instance, after long debate

quantum bit commitment protocols were proved to be insecure against simple tricks of

entanglement attacks [119, 125]. Furthermore, some theoretical proposals of QIP require

stringent resources that could not be achieved by technologies in foreseeable future. All of

2


these software issues hinder QIP applications from being realised and practically applied.

To improve the practicality of a QIP application, both the hardware and software

issues have to be settled. This can be done by exploring possible implementation flaws

and invalid theoretical assumptions that would affect the functionality of the applica-

tion, modifying the device architecture and the protocol procedure in order to improve

the application’s efficiency and feasibility, and proposing new QIP applications that are

practically useful and could be implemented with near future technology.

In my PhD research, I particularly study two of the most mature QIP applications:

quantum cryptography, which has already been commercialised and elementary real-life

applications have been demonstrated [1]; and the ion trap quantum computer, which

is widely recognised as a promising implementation [182] that holds several records of

quantum computation (see Ch. 4 for more details). Most of the results of my work is

included in this thesis. The abstract of each chapter is presented as follows:

In Chapter 2, I present my work with Hoi-Kwong Lo about Position-Based Quan-

tum Cryptography (PBQC). PBQC aims to send a secret message to a person that the

security of the message is guaranteed by the person’s geographical location. Since the

concept of PBQC was developed as early as in 2002 [96], it was generally believed to

be unconditionally secure. However, we discovered a loophole that allows the dishonest

parties to obtain the exact secret message if they possess entangled quantum resources.

We found that all PBQC protocols, at the time our work was conducted, were insecure

against our attack. We also calculated the minimum amount of entanglement required

for a hacking. Based on the idea of our work, PBQC was later shown to be generally

insecure [43]. The result of this work was presented by Hoi-Kwong Lo in the CIFAR QIP

meeting 2010 1, and was published as Ref. [108].

In Chapter 3, I present my work with Christian Weedbrook about quantum secret

sharing (QSS). We studied the performance of a QSS scheme when a continuous-variable

cluster state is employed as the resources. In the literature, the security of continuous

variable QSS is guaranteed only in the scenario that the quantum state involves infinite

energy, which is unrealistic. In our work, we quantified the amount of leaked secret infor-

mation when finite energy states are employed. We also developed strategies to protect

the security of QSS against such information leakage. Our results relax the stringent

requirement of resources that can be used for secure QSS. Our work was presented in the

APS DAMOP meeting 2013 2, and was published as Ref. [109].

1‘Position-based quantum cryptography: Efficiency of cheating strategies’, Canadian Institute forAdvanced Research (CIFAR) Quantum Information Processing meeting, Toronto, November 2010

2‘Quantum secret sharing with continuous variable cluster states’, American Physical Society (APS)Division of Atomic, Molecular, and Optical Physics (DAMOP) Meeting 2013, Quebec City, Canada,

3


In Chapter 4, I introduce the background of the ion trap system that I studied, and

the mathematical techniques that I employed to describe the dynamics of trapped ions.

In Chapter 5, I present my work with Daniel James in examining the speed limit of

a scalable ion trap quantum computer. The quantum information stored in ions would

be influenced by direct current (dc) Stark effect if the ion transportation is too fast.

We formulated the relation between the transportation speed and the magnitude of the

Stark effect. Then we suggested an optimised transportation trajectory that minimises

the influence. From the results, we defined a threshold speed of ion transportation, above

which dc Stark effect would significantly alter the encoded quantum information. Our

work was presented as a poster in the SQuInT meeting 2011 3, and was published as

Ref. [106].

In Chapters 6 and 7, I present my work with Daniel James in proposing a new QIP

application for the ion trap system: a universal bosonic simulator (UBS). The idea is that

the motional state of trapped ions exhibits bosonic behaviours, so the system can be used

to study the physics of other bosonic systems, such as optical systems. When comparing

to conducting experiments directly on optical systems, our UBS has the advantages that

state preparation can be more flexible, measurements can be more accurate, and nonlinear

bosonic interaction is tuneable and can be arbitrarily strong.

We have proposed two trapped ion UBS architectures. The architecture in Chapter 6

involves multiple ions trapped in a single harmonic potential. The initialisation and

transformation of bosonic states can be conducted by applying radiation fields with

precisely tuned frequencies. Although the quality of operation is substantially reduced

when more than 4 bosonic modes are simulated, this architecture can be realised with

present technology, and thus be useful for demonstrating simple but important bosonic

phenomena. This work was presented as a poster in the ICAP meeting 2012 4.

In Chapter 7, I introduce another UBS architecture that involves separately trapped

ions. The initialisation and transformation of bosonic states can be achieved by varying

the trap potential. This architecture is more scalable because the accuracy of mode

operations will not reduce as the scale of simulation increases. The result of this chapter

was also presented as a poster in the ICAP meeting 2012, and was published as Ref. [107].

In Chapter 8, I present my own work about a new method to re-cool ions during

quantum computation. Currently, ion re-cooling process is considered as the speed bot-

June 20133‘Dephasing of trapped-ion qubit due to Stark shift during shuttling’, 13th annual Southwest Quan-

tum Information and Technology (SQuInT) meeting, Boulder, U.S.A., February 20114‘Proposal for ion trap bosonic simulator’, International Conference on Atomic Physics (ICAP) 2012,

Palaiseau, France, July 2012

4


tleneck of an ion trap quantum computer. Re-cooling ions by state-of-the-art laser cooling

takes an order of magnitude longer time than other quantum logical operations [78]. My

method could resolve this speed bottleneck because laser cooling is not involved during

the quantum computation. The principle of my method is to first prepare some coolant

ions in the ground state. When a qubit ion has to be re-cooled, it collides with a coolant

ion. I show that if the collision is well-controlled, the motional excitation of the qubit

ion will be completely transferred to the coolant ion. By using this method, a qubit can

be re-cooled ten times faster than by laser cooling, thus the clock rate of an ion trap

quantum computer can be improved. The result of this work was presented in the APS

March meeting 2013 5, and was submitted to the Physical Review A [105].

In Chapter 9, I summarise my thesis and discuss several possible research directions

in the future.

5‘Rapid laser-free ion cooling by controlled collision’, American Physical Society March Meeting 2013,Baltimore, U.S.A., March 2013

5

Chapter 2

Position-Based Quantum

Cryptography

2.1 Introduction

In everyday life, we constantly place trust on spatial locations. For instance, when we

deposit money in a bank, we seldom request the teller to prove his/her identity as a bank

employee. This is because we believe the area behind the bank counter is a secure region

where an imposter is rather difficult to get into, and more importantly, we have verified

by our eyes that the “teller” is indeed inside this secure region.

However, there are more cases that we cannot verify a person’s spatial location directly

by our eyes. For example, if a postman is delivering a registered parcel, the recipient

may be located in a distant building of where our vision is blocked by concrete walls. At

this time, we may want a method to verify if the postman is delivering the parcel to the

location we want.

Location verification is believed to be achievable by a class of informatics method

referred as position-based cryptography (PBC) [42]. The principle of PBC is to set up

several trusted reference stations, called verifiers, to send out messages to the person,

called prover, who is supposed to be in a designated position. In an ideal PBC protocol,

the messages contain a challenge that the correct response can be made by only the

prover at (or in the small neighbourhood of) the designated location. Thus the prover’s

credential of spatial location can be verified by making the correct response.

PBC was first studied in classical settings, of which the challenges are prepared as

classical messages. Unfortunately, unconditionally secure 1 classical PBC has been proven

1Here I define the unconditional security as: no matter what kind of resources and cheating strategiesare employed by the dishonest parties, they cannot obtain any information about the secret message

6

Chapter 2. Position-Based Quantum Cryptography

to be impossible [42, 43]. More explicitly, for any classical challenges sent by the verifiers,

a group of cheaters, none of whom is inside the secure location, can reproduce the exact

response as the prover. The main problem of the classical PBC was once (in 2010)

believed to be that classical challenges can be cloned and re-sent by the cheaters.

If the problem is the duplicability of classical messages, a natural improvement of

PBC is to employ quantum messages as the challenges. It is a well-known property

that an arbitrary quantum state cannot be deterministically cloned with unity fidelity

[185, 52]. Based on this property, the quantum extension of some cryptographic tasks

can be unconditionally secure, such as quantum key distribution [26, 60, 126, 120, 162]

and quantum secret sharing [83, 51]. The possibility of Position-Based Quantum Cryp-

tography (PBQC) was first studied by Kent under the name of ‘quantum tagging’ as

early as in 2002 [96], and the idea was later revisited independently by Chandran et al.

[43] and Malaney [123, 122] at around 2010.

In the contrary to the authors’ claims of unconditional security, both the PBQC

protocols suggested by Chandran et al. and Malaney were in fact insecure. As discussed

by Kent, Munro, and Spiller [97], and in our work [108] (our publication that contains the

material of the current chapter), cheaters can use entanglement resources and non-local

quantum operations to produce the same response as if the prover in the secure region.

By generalising the cheating strategy in [97, 108], every PBQC protocol was later shown

to be insecure [39] and can be cheated efficiently [23].

In this chapter, I summarise my work in 2010 about the security of PBQC, which the

material was published as Ref. [108]. In Sec. 2.2, I introduce the idea of position-based

cryptography and discuss the insecurity of the classical protocols. In Sec. 2.3, I review

the PBQC protocols in Refs. [39, 123], which are all known protocols when my work

was conducted. In Sec. 2.4, I present the cheating strategy for the case that the verifiers

and the prover are collinear located. In Section 2.5, I present the cheating strategy for a

more general case that the verifiers are distributed in three-dimensional space. Discussion

about the insecurity of the PBQC protocols and the loopholes of the claimed security

proof are discussed in Sec. 2.6. In Sec. 2.7, I outline a modified protocol, of which the

security analysis is given in Sec. A.1. I conclude this chapter in Section 2.8 with a remark

about the prospect of PBQC.

without being detected by the honest parties.

7


2.2 Classical Position-Based Cryptrography

To appreciate the motivation of PBQC, it would be better to first understand the idea of

classical PBC. For simplicity, I assume that all parties have synchronized clocks and work

with a flat Minkowski space-time. A prover is supposed to be located at the position

P surrounded by a finite secure region, to where no cheaters can access. To conduct

a PBC protocol, N verifiers are established, at locations V1, . . . , VN , around P . For

simplicity, I hereafter assume P is equidistant to all verifiers, but PBC can be modified

to incorporate non-equidistant verifiers by changing the sending time of the messages.

The simplest implementation of a PBC protocol involves two verifiers that are collinear

with the secure region, and the prover is supposed to be in the middle between the

verifiers. The layout is shown in Fig. 2.1 when t = 0.

Before the PBC starts, the verifiers have to decide a challenge for the prover, and the

information of the challenge is divided and distributed to each verifier. In the classical

case, the challenge can be a secret encoded by classical secret sharing [31, 160]. As an

example, the challenge is that the prover has to report a number, which is the sum of

the numbers possessed by each verifier.

When PBC starts, the divided information is sent simultaneously from the verifiers.

The functionality of PBC is based on the fact that information cannot travel faster than

the speed of light. As shown in the space-time diagram in Fig. 2.1, the signals are

gathered at the soonest when the prover is located at the mid-point between the verifiers

(point P in Fig. 2.1). Then the prover immediately computes the answer of the challenge,

and reports the answer 2. In the original PBC [42], the prover’s location can be verified

by the time that the response reaches the verifiers. In the honest case, i.e., the prover

is located at the mid-point, response is made at t = d/c and will reach the verifiers at

t = 2d/c.

I note that although the verifiers are collinear for N = 2, or coplanar for N = 3, all

three coordinates of the prover can be verified. This is because the prover can take the

shortest time to make the response, only if he is located at the mid-point of the straight

line (for N = 2) or the centre of the triangle (for N = 3). I also note that P should be

surrounded by the verifiers, even if each verifier can choose a different time of sending

message. Otherwise, there must be places surrounded by V1, . . . , VN such that shorter or

equal time is needed to receive all information. The idea is illustrated on Fig. 2.2.

2Unless specified, I assume hereafter that the operation time of the detectors, the computationaldevices, and the emitters are negligible when comparing with the travel time of the challenge.

8


B1 B2 V2V1 P

t t

(d-l)/c

(d+l)/c

2d/c

0

2d

2l

Figure 2.1: Space-time diagram of the one dimensional position-based cryptography sce-nario. The verifiers are separated by a distance 2d, and the secure region (shaded area) isin the middle with length 2l. Solid lines denote space-time trajectory of the information,which can be quantum or classical, while double lines denote that of classical informationonly. The protocol starts at t = 0, when message of the challenge is sent by the verifiers.At t = (d − l)/c (squares), the messages reach the boundary of the secure region, theclosest to the mid-point the cheaters can locate. At t = (d − l)/c (circles), the messageor response exits the secure region. In the honest case, i.e., the prover is located at themid-point, response is made at t = d/c and reach the verifiers at t = 2d/c. If there isonly one cheater at B1, the correct response is made at t = (d + l)/c and reach V2 attime later than 2d/c (trajectory is shown as dot-dashed line).

9


V1

V2V3

P

P2

Figure 2.2: At a particular time t1, the front of signals sent from V1, V2, V3 are representedby solid, dashed, and short-dashed lines respectively. While signals reach P at t = t1,another position P2 inside the triangle of three verifiers (framed by dotted lines) canobtain all information before t1.

2.2.1 Individually Cheating

The classical PBC is secure if there is only one cheater. In the case shown in Fig. 2.1, let

us assume the cheater is located at B1. The cheater can intercept the message from V1 at

t = (d− l)/c, but the message from V2 reaches at t = (d+ l)/c at the earliest. Therefore,

the response can be made only at t = (d + l)/c, and the response will reach V2 at time

2l/c later than in the honest case. Thus the verifiers can detect the presence of cheater.

2.2.2 Collaborative Cheating

However, the classical PBC is insecure if more than one collaborating cheaters are in-

volved. In the case shown in Fig. 2.1, one cheater is located at B1 and B2. At t = (d−l)/c,the cheater at B1 (B2) intercepts the message from V1 (V2). A copy of the message is

saved before it is forwarded to another cheater. At t = (d + l)/c, both cheaters gather

both piece of message. Therefore the correct response can be made, and the response

will reach the verifiers at the same time as if produced by the prover inside the secure

region.

2.3 PBQC Protocols

The crucial step in the collaborative cheating of classical PBC is to faithfully clone

the message from the verifier. The aim of PBQC is to solve this problem by using the

quantum property of no-cloning. While the layout and procedure of PBQC is the same as

classical PBC, the only difference is that the verifiers encodes the challenge information

in a (or part of an entangled) quantum state, and the challenge for the prover is a

10


transformation or measurement of the state. The information to be reported can be

the transformed quantum state or the measurement result. In the following, I describe

the protocols suggested by Chandran et al. [43] (hereafter referred as Protocol A) and

Malaney [123, 122] (hereafter referred as Protocol B).

2.3.1 Protocol A

The idea of Protocol A is to send the basis of measurement and the encoded qubit

separately from different verifiers. Security of this protocol relies on the idea that a

quantum state can be measured perfectly (obtain the information encoded in a quantum

state deterministically with 100% accuracy) only if the correct measurement basis is

known. The procedures of Protocol A is shown as follows.

Step 1. The verifier at V1 encodes a classical message u ∈ 0, 1 as a qubit |u〉, where|0〉 and |1〉 are respectively the +1 and −1 eigenstate of the Pauli Z operator 3. Inspired

by the BB84 QKD protocol [26], the message is encrypted by applying the transformation

Hq on the qubit, where H is the Hadamard gate 4, and q is a random bit valued 0 or 1.

Step 2. The V1 verifier generates N − 2 random bits q2, q3, . . . , qN−1, and decide a bit

qN by the relation

q = q2 + q3 + · · ·+ qN mod 2 . (2.1)

The bits q2, q3, . . . , qN are respectively distributed to the verifiers at V2, . . . , VN . The

classical message u is also sent to other verifiers. The communication between verifiers

is assumed to be secure, for example QKD system is employed.

Step 3. Verifiers compromise to send the messages to prover at the same time t = t0.

The V1 verifier sends the encoded qubit Hq|u〉, while other verifiers send the classical bits

qi.

Step 4. Upon receiving all information, the prover at P adds up all classical bits

to obtain q. The qubit can then be decrypted by applying Hq. The decrypted state is

measured in Z basis to obtain the encoded message. The prover immediately reports the

results to all verifiers.

Step 5. If qi’s are random, missing any one classical bit would cause half chance of

wrong measurement basis. verifiers validate the identity of the prover by checking if the

reported message matches the encoded message. By checking the arrival time of the

response, the location of the prover is also verified.

3Throughout this chapter, the eigenvalues and measurement outcomes of a Pauli operator are +1 or-1.

4i.e., H |0〉 = |+〉 and H2 = 1, see Ref. [143] for more information

11


2.3.2 Protocol B

The idea of Protocol B [123] is to encode information into a maximally entangled state.

The state is then distributed to verifiers, and encrypted by each verifier with a random

local transformation. The classical information about the transformation will be sent

from the verifiers. Security of this protocol relies on the intuition that the state cannot

be decrypted and then perfectly measured before all local transformation information is

gathered. The procedure of Protocol B is shown as follows.

Step 1. N bits of message is encoded as aN qubit Greenberger-Horne-Zeilinger (GHZ)

state

|GHZ〉 = 1√2(|a1〉|a2〉 . . . |aN〉 ± |1⊕ a1〉|1⊕ a2〉 . . . |1⊕ aN〉) , (2.2)

where a1, . . . , aN ∈ 1, 0; ⊕ denotes addition with modulus 2. Each qubit of the state

is distributed to a separate verifier.

Step 2. After encrypted by a local transformation Ui, the verifiers send their qubit to

the prover. The prover will store the entangled state in a perfect quantum memory.

Step 3. At an agreed time t = t0, each verifier sends the classical information about

the transformation Ui to the prover.

Step 4. After the classical information is received, the prover decrypts the state

back to a N -qubit GHZ state. The encoded message is then obtained by quantum

measurement. The result is immediately announced.

Step 5. The measurement result is probably wrong if the state is measured before

decryption. Hence the identity of the prover can be authenticated from the announced

result, and the prover’s location can be verified by the time of response.

2.4 Cheating in N = 2 Case

Contrary to the claim(s) of unconditional security, both Protocols A and B are, in fact,

insecure. I first demonstrate the cheating strategy for the case with two verifiers. I will

consider the scenario shown in Fig. 2.1. The general strategy for the case with more

verifiers will be discussed in the next section.

2.4.1 Cheating against Protocol A

Protocol A was once believed to be unconditional secure. In fact, a detailed claim of

security based on complementary information tradeoff was given in Ref. [43]. The

intuition behind the claimed security proof is that, if the correct basis is not known, any

12


measurement on the encrypted qubit would inevitably disturb the state. Therefore the

measurement outcome would be wrong with non-zero probability.

The problem of this claim is the implicit assumption that no prior entanglement is

shared by the cheaters. In fact, quantum teleportation can be conducted by appropri-

ately measuring the qubit and the shared entangled resources [27]. The teleportation

measurement would not extract any information about the quantum state. Thus the

security claim by Chandran et al. is wrong in this case, i.e., measurement can cause no

distortion on the qubit, while the no-cloning theorem is still obeyed. The main idea of

my cheating scheme in the N = 2 case is to teleport the encrypted qubit from B1 to B2

for measurement in the correct basis. Detailed procedure of the cheating strategy is as

follows.

Step 1. Before the cheaters move to the destination, they come together and each

pick a qubit from a Bell state

|Φ00〉 ≡1√2(|00〉+ |11〉) = 1√

2(|++〉+ | − −〉) = 1√

2(|+i −i〉+ | −i +i〉) . (2.3)

5 Their quantum memory is assumed to be perfect, so that the state remains coherent

until measurement.

Step 2. At t = 0, V1 verifier sends out the qubit Hq|u〉, and V2 verifier sends out the

classical bit q2 = q that contains the basis information. At t = (d − l)/c, B1 cheater

captures the qubit and B2 cheater obtains the classical bit. To avoid suspicion of the

prover, I assume the cheaters have the power to send dummy qubit and basis information

to the prover, and block the prover’s subsequent response. I hereafter neglect the role of

the prover in the cheating procedure.

Step 3. B1 cheater immediately teleports the captured qubit to B2 cheater by per-

forming a Bell measurement on the captured and the Bell state qubit. The circuit of

the Bell measurement is given in Fig. 2.3. The measurement outcomes of the encrypted

qubit, s1, and Bell state qubit, s2, are then sent to B2 cheater.

Step 4.At the same instance t = (d − l)/c, the teleported qubit being held by B2

cheater becomes

X(1−s2)/2Z(1−s1)/2Hq|u〉 . (2.4)

Let us consider if q = 0, the state becomes

X(1−s2)/2Z(1−s1)/2|u〉 = (−1)u(1−s1)/2|u⊕ (1− s2)/2〉 . (2.5)

5|±i〉 = (|0〉 ± i|1〉)/√2 is the ±1 eigenstate of the Y operator.

13


X

Z

Figure 2.3: Circuit for teleporting an unknown qubit |ψ〉 [143, 27]. Measurement isdenoted as squares, the measurement basis is represented by the character inside thesquares. Because no-signalling theorem has to be obeyed, the teleported state is trans-formed by a byproduct, UΣ = X(1−s2)/2Z(1−s1)/2, which depends on random measurementoutcome s1 and s2.

Since B2 cheater knows the basis is Z, and the state in Eq. (2.5) is an eigenstate of the

Pauli Z operator, a perfect measurement can be conducted with the outcome (−1)us2.

On the other hand, if q = 1, the state becomes

HZ(1−s2)/2X(1−s1)/2|u〉 = (−1)[u⊕(1−s1)/2](1−s2)/2H|u⊕ (1− s1)/2〉 . (2.6)

Since B2 cheater knows the basis is X , and the state in Eq. (2.6) is an eigenstate of the

Pauli X operator, a perfect measurement can be conducted with the outcome (−1)us1.

B2 cheater immediately sends the measurement outcome to B1 cheater. I note that al-

though the measurement outcome of B2 cheater contains information about the outcome

of B1 cheater, no superluminal communication can be implemented by using quantum

teleportation. This is because B1 cheater cannot choose the measurement result deter-

ministically.

Step 5. At t = (d+ l)/c, both B1 and B2 cheaters know all the measurement results,

i.e., s1, s2, (−1)us1 or (−1)us2, as well as the correct measurement basis, q. The secret

u can then be inverted by both cheaters. B1 (B2) cheater reports the result to V1 (V2)

verifier. As shown in Fig. 2.1, both verifiers will receive the correct response at t = 2d/c.

The cheaters use the same amount of time to produce the same correct response as

the prover, Protocol A is therefore insecure.

The crucial process in the cheating strategy is the teleportation in Step 3. The

teleported state received by B2 cheater will be acted upon by one of the four teleportation

byproduct, I,X, Z and XZ. Since the encrypted state is an eigenstate of either X and

Z, the teleported state would either be orthogonal to or the same as (up to an irrelevant

overall phase) the encrypted state. In other words, the teleportation does not change

the basis of the state; this is the reason that the cheating strategy works. Therefore,

14


B2 cheater, who has the basis information, can simply measure the qubit in that basis

without disturbing the state. Subsequently, after hearing the actual Bell measurement

outcomes from B1 cheater, B2 cheater will be able to tell what the encrypted state is.

For this reason, cheating can be successful with certainty.

The cheating strategy can also be understood in another viewpoint. Since the mea-

surements by B1 and B2 cheaters commute, it is also legitimate to consider that B2

cheater performs the measurement before B1 cheater does. In this case, B2 cheater mea-

sures the Bell state qubit in the correct basis. By the Einstein-Podolsky-Rosen effect,

the Bell state qubit being held by B1 cheater will be projected to either the same state

or the opposite state to the encrypted qubit. Then the duty of B1 cheater is to perform

a parity check on the two qubits.

While a parity check is impossible for arbitrary basis, it is possible for the basis only

considered by Protocol A: Pauli X and Z basis. In this case, it happens that the operator

XX commutes with ZZ, so B1 cheater can perform a parity check by simply doing a

Bell measurement. More explicitly, the Bell measurement will project the qubits to one

of the four states in Eq. (2.3) and

|Φ01〉 ≡ 1√2(|00〉 − |11〉) = 1√

2(|+−〉 + | −+〉) = 1√

2(|+i +i〉+ | −i −i〉) , (2.7)

|Φ10〉 ≡ 1√2(|01〉+ |10〉) = 1√

2(|++〉 − | − −〉) = −i√

2(|+i +i〉+ | −i −i〉) , (2.8)

|Φ11〉 ≡ 1√2(|01〉 − |10〉) = −1√

2(|+−〉 − | −+〉) = i√

2(|+i −i〉 − | −i +i〉) . (2.9)

If the qubits are in the Z basis and same (different) in parity, the measurement results

can be only |Φ00〉 and |Φ01〉 (|Φ10〉 and |Φ11〉). Similarly, if the qubits are in the X basis

and same (different) in parity, the measurement results can be only |Φ00〉 and |Φ10〉 (|Φ01〉and |Φ11〉) 6. For these reasons, Protocol A can be cheated successfully with certainty.

2.4.2 Cheating against Protocol B

In the two-verifier case, 2 bits of information, ab = 00, 01, 10, 11, are encoded into one

of the four Bell states |Φab〉 [123], i.e., Eqs. (2.3) and (2.7)-(2.9). After encrypted by

random local transformation U1 and U2, the state is sent to P . At t = 0, V1 (V2) verifier

broadcasts U1 (U2). If the prover is at P , correct response will reach verifiers at t = 2d/c.

To cheat this protocol, each cheater captures and decrypts a qubit when the decryp-

tion information is received. The decrypted state is then teleported to one cheater for

6I note that Bell measurement can also be checked the parity for two qubits in Y basis.

15


Bell state measurement. The step by step procedure is shown as follows.

Step 1. Before the PBQC process, the cheaters share a Bell state as in Eq. (2.3), and

store it in good quantum memory.

Step 2. The cheaters capture the state sent by the verifiers. The state is then stored

in another quantum memory.

Step 3. At t = (d− l)/c, B1 (B2) cheater receive the classical information of U1 (U2)

from the V1 (V2) verifier. The stored state can then be decrypted by the inverse of the

local transformation, i.e., U †1 and U †

2 . The encoded state |Φab〉 is hence recovered.

Step 4. Immediately after the decryption, B2 cheater teleports the encoding qubit to

B1 cheater by using the pre-shared Bell state. To explicitly describe the change of the

state during the teleportation, I here employ the stabilizer formalism as introduced in

Ref. [2]. Before the teleportation, the stabilisers of the states are

K1 = (−1)aZ1Z2 ; K2 = (−1)bX1X2 ; K3 = Z3Z4 ; K4 = X3X4 , (2.10)

where qubits 1 and 2 are respectively the encoded qubit captured by B1 and B2 cheater;

qubits 3 and 4 are respectively the pre-shared Bell state qubit of B1 and B2 cheater. B2

cheater first applys a CNOT gate on his qubits, the stabilisers then become

K1 = (−1)aZ1Z2 ; K2 = (−1)bX1X2X4 ; K3 = Z2Z3Z4 ; K4 = X3X4 . (2.11)

Qubit 2 is then measured in the X basis and the result is s2; qubit 4 is measured

in the Z basis and the result is s4. These outcomes are then sent to B1 cheater. The

stabilizers after the measurement is

K ′1 = (−1)as4Z1Z3 ; K ′

2 = (−1)bs2X1X3 ; K ′3 = s2Z2 ; K ′

4 = s4X4 . (2.12)

Qubit 2 and 4 are obviously no longer entangled as they are measured. K ′1 and K ′

2

show that qubits 1 and 3 become a Bell state |Φa′b′〉, where a′ = a + (1 − s3)/2 and

b′ = b+ (1− s2)/2. So B1 cheater can measure the state perfectly by Bell measurement,

the outcomes a′ and b′ are then sent to B2.

Step 5. At t = (d+ l)/c, both cheaters obtain information from each others. The en-

coded message a and b can be calculated from a′, b′, s2 and s3. The cheaters immediately

report the result, which will reach the verifiers at t = 2d/c. Since the cheaters can make

the correct response within the same time as the prover, Protocol B is thus insecure.

16


d

l

V1

V2V3

B2

B1

B3

P

Figure 2.4: Locations of the three verifiers, three cheaters, and the prover are shown asblack dots. The shaded region represents the restricted area surrounding P . Withoutcheating, information flows along solid lines; if cheating presents, information flows alongsolid lines outside the restricted area and follows dotted lines in the restricted area. Asthe path of V2 → P → V1 is longer than V2 → B2 → B1 → V1, the process of cheatingcosts shorter time than the honest case.

2.5 Cheating in N > 2 Case

The above cheating strategy can be generalised to the situation with more than two

verifiers. I first consider the case with three verifiers; the case with more verifiers will

be discussed later. For simplicity, I assume the location of verifiers, V1, V2, V3, are the

vertices of an equilateral triangle. The location of the prover, P , is the centre of the

triangle, with a distance d from each verifier. P is surrounded by a restricted area with

radius l. In this case, three cheaters are sufficient to cheat both Protocol A and B. I

assume the cheaters are located at B1, B2, B3, which are just outside the restricted area

and along the straight lines linking P with V1, V2, V3. The layout of the scenario is shown

in Fig. 2.4.

2.5.1 Cheating against Protocol A

In this protocol, V1 verifier prepares the encoded state |u〉 and encrypts it as Hq|u〉. Theencrypting information q2 and q3, where q = q2 + q3 is distributed to the verifiers at V2

and V3 respectively. At t = 0, the V1 verifier sends the encrypted state, and V2 and V3

cheaters send q2 and q3, to the prover. They expect the result of u will return at t = 2d/c.

In this case, the cheaters are not going to teleport the qubit as there is only one

qubit but two separate pieces of encrypting information. Instead they need a method to

share the encrypting information. I find this task can be accomplished by techniques in

measurement-based quantum computation (MBQC) [147, 148]. The steps of the cheating

17


Table 2.1: Tables of stabilizers in different cases of qi’s. K1 is the stabilizer of GHZstate compatible with the measurement basis. K ′′

1 , K′′2 , and K

′′3 are stabilizers after the

measurement according to the cheating scheme.

q2 q3 K1 K ′′2 K ′′

3 K ′′1

0 0 X1X2X3 s2X2 s3X3 s2s3X1

0 1 −Y1X2Y3 s2X2 s3Y3 −s2s3Y11 0 −Y1Y2X3 s2Y2 s3X3 −s2s3Y11 1 −X1Y2Y3 s2Y2 s3Y3 −s2s3X1

scheme are shown as follows.

Step 1. Before the PBQC process, each cheater share a qubit from a 3-particle GHZ

state, i.e,

|Φ000〉 =1√2(|000〉+ |111〉) . (2.13)

Step 2. The verifiers send out their information at t = 0. At t = (d− l)/c, B1 cheater

captures the encrypted state, and B2 and B3 cheaters get q2 and q3.

If qi = 0, Bi cheater measures the GHZ qubit in X basis; otherwise he measures in Y

basis. As a property of the GHZ state, if both B2 and B3 cheater measure in the same

basis, the GHZ qubit holding by B1 cheater becomes an eigenstate of Pauli X operator,

otherwise it is an eigenstate of Y operator. To explain this property more explicitly, let

us consider the stabilisers of the GHZ state in Eq. (2.13) are

K1 = X1X2X3 ; K2 = Z1Z2 ; K3 = Z1Z3 . (2.14)

When qubit 2 (3) is measured in Y , the stabiliser K1 has to be modified by multiplying

with K2 (K3). The resultant stabilisers in each case is shown in Table 2.1. After measure-

ment, the compatible operator in a stabiliser is replaced by a number representing the

measurement result 7. Upon all the combination of q2 and q3, qubit 1 will be stabilised

by X (Y ) if qubit 2 and 3 are measured in the same (different) basis.

Step 3. Immediately after the measurement, B1 cheater applies a Hadamard trans-

formation H followed by a π/4-gate, S 8, on the GHZ state qubit. This operation is to

transform the eigenstates of X to that of Z, and the eigenstates of Y to that of X , with

the same eigenvalues. After this operation, the GHZ qubit of B1 cheater will become a

Z eigenstate if q2 + q3 is even, otherwise the qubit is a X eigenstate. At the the same

7More information about the stabiliser formalism is referred to Refs. [143] and [2].8S|+〉 = |+i〉; S2 = Z.

18


time t = (d− l)/c, B1 cheater also receives the encrypted qubit. It is easy to see that the

two qubits of B1 cheater are in the same basis, i.e. both qubits are eigenstates of either

X or Z. B1 cheater then performs a parity check by conducting a Bell measurement.

Step 4. The cheaters share their measurement outcomes and basis information. Since

the mutual distance between B1, B2 and B3 is√3l, the cheaters can obtain all the

information at t = (d + (√3 − 1)l)/c. From the information of B2 and B3 cheaters,

the actual state of the GHZ qubit of B1 cheater is known from Table 2.1. With the

information of parity from the Bell measurement outcome, the state of the encoded

qubit is obtained.

Step 5. If the correct result is immediately reported by the cheaters at t = (d+(√3−

1)l)/c, the response will reach the verifiers at t = (2d+(√3−2)l)/c, which is earlier than

the expected response from the prover at t = 2d/c. The cheaters can simply delay the

report for an appropriate time, in order to match the time span in honest case. Hence

the protocol is cheated.

I note that the time is shortened because information takes 2l/c time to travel from

B1 to B2 in the honest case, while only√3l/c is needed if there are cheaters. In general if

the cheater’s locations do not form an equilateral triangle, the cheating scheme may still

process faster than the honest case, provided that P is not on the same straight line as

any two verifiers. This is because in the honest case, the information has to be sent from

a vertex to the centre of triangle, and then resent to another vertex, while information

from the cheaters can be sent through a shortcut, as shown in Fig. 2.4.

The above cheating scheme can be generalized to cases with N > 3 verifiers; at most

N cheaters are needed in each case. Before the PBQC process, the cheaters create a N

particle GHZ state, of which the stabilisers are

K1 = X1X2 . . .XN , Ki = Zi−1Zi , (2.15)

for i = 2, 3, . . . , N . The ith cheater picks the ith qubit from the GHZ state, and travels

to a position Bi between P and Vi.

When each cheater at B2, . . . , BN receives the basis information, the GHZ qubit is

measured in X basis if qi = 0, or Y basis if qi = 1. The result is sent to other cheaters. If

even number of q’s are equal to 1, the qubit of B1 is will be in the X basis, otherwise it be-

comes an eigenstate of the Y basis. Such phenomenon can be understood as the following.

In the even number case, the Y measuring qubits can be grouped in pairs. Let us consider,

for example, the first pair is the qubits m and n. In order to construct a stabiliser that

19


is compatible to the measurement basis 9, the first stabiliser is modified by multiplying

those in Eq. (2.15) as K ′1 = K1Km+1Km+2 . . .Kn = −X1 . . . YmXm+1 . . .Xn−1YnXn+1 . . . .

This new stabiliser K ′1 is compatible to the measurement basis of the qubits m and n.

The stabiliser can be similarly modified to make compatible to other pairs of Y mea-

suring qubits. The final stabiliser will consist of a X1 operator, Y operator for those Y

measuring qubits, and X operator for the X measuring qubits. When all qubits except

qubit 1 are measured, qubit 1 will then become an eigenstate of X , with an eigenvalue

depending on measurement outcomes of other qubits.

On the other hand in the odd number case, the Y measuring qubits can also be

grouped in pair with one singled out. The stabiliser can be modified as the even number

case for the pairs. For the singled out qubit, say qubit r, the stabiliser has to be modified

as K ′1 = K1K2 . . .Kr = −Y1 . . . Yr . . . . Then the final stabiliser will consist of a Y1

operator, Y operator for those Y measuring qubits, and X operator for the X measuring

qubits. When all qubits except qubit 1 are measured, qubit 1 will then become an

eigenstate of Y , with an eigenvalue depending on measurement outcomes of other qubits.

Identical to the N = 3 case, B1 cheater applies a SH gate onto his GHZ state qubit,

he then obtains an eigenstate of X operator if q is odd, or an eigenstate of Z operator if

q is even. He then check the parity of the GHZ state qubit and the encrypted qubit by

measuring in the Bell basis; the measurement outcome is then shared with other cheaters.

In the present case of N > 3, cheaters do not receive all information at the same time,

but it is easy to check that even the slowest piece of information should arrive at the

latest as in the honest case. The information provided by B2, . . . , BN cheaters determines

the actual state of the GHZ state qubit of B1 cheater, and the parity check measurement

of B1 cheater reveals the identity of the encrypted qubit. Hence the value of encoded

message is obtained and sent to the verifiers by the cheaters. The whole process takes

time fewer than or the same as in the honest case.

2.5.2 Cheating against Protocol B

In Protocol B with three verifiers, three bits of information is encoded as one of the eight

tripartite GHZ states [123] characterized by parameters b1, b2, b3:

|Φb1b2b3〉 =1√2(|0〉|b2〉|b3〉+ (−1)b1 |1〉|1⊕ b2〉|1⊕ b3〉) , (2.16)

9If a stabiliser is compatible to a measurement basis, it is a tensor product of the measurementoperator.

20


where b1, b2, b3 ∈ 0, 1. This state is stabilised by

K1 = (−1)b1X1X2X3 ; K2 = (−1)b2Z1Z2 ; K3 = (−1)b3Z1Z3 . (2.17)

The ith GHZ qubit is distributed to the Vi verifier. The qubit is then encrypted by an

arbitrary local transformation Ui, and all encrypted qubits are subsequently sent to P .

At t = 0, verifiers send classical information of Ui to P . In the honest case, the prover

decrypts and measures the state to get b1, b2, b3. Correct results should return to the

verifiers at t = 2d/c.

In this case, three cheaters are sufficient to cheat. The idea is the same as in the

N = 2 case. The cheaters decrypt and teleport the captured qubits to one cheater. Then

the encoded information can be obtained by conducting the GHZ state measurement.

The cheating strategy is as follows.

Step 1. Before the PBQC process, a Bell state in Eq. (2.3) is shared between B1 and

B2 cheaters, as well as between B1 and B3 cheaters. I denote the Bell state qubits of B1

and B2 (B1 and B3) cheaters as qubit 4 and 5 (6 and 7).

Step 2. The cheaters capture the encrypted qubits. At t = (d− l)/c, cheaters receive

the classical information of Ui. The qubits are then decrypted and the encoding state is

obtained.

Step 3. B2 and B3 cheaters teleport the encoded qubits by using the teleportation

protocol described in Fig. 2.3. Afterwards, the qubits of B1 cheater is stabilized by

K ′1 = (−1)b1s2s3X1X4X6 ; K ′

2 = (−1)b2s5Z1Z4 ; K ′3 = (−1)b3s7Z1Z6 . (2.18)

These stabilisers indicate that qubits 1, 4 and 6 become a GHZ state |Φb′1b′2b

′3〉, where

b′1 = b1+(1−s2s3)/2, b′2 = b2+(1−s5)/2, b′3 = b3+(1−s7)/2. B1 cheater then conducts

a GHZ state measurement to reveal b′1, b′2, and b

′3. The result is sent to other cheaters.

Step 4. Information exchange among cheaters is finished at t = (d + (√3 − 1)l)/c.

The encoded message b1, b2, b3 can easily be inferred from the measuring outcomes. Just

as the cheating of Protocol A, the cheaters delay the report for an appropriate time, so

the results can reach the verifiers at t = 2d/c. Since the response is the same as if the

prover is at P , Protocol B is hence cheated.

For N > 3, the procedure of cheating is more or less the same. Each cheater shares

a Bell state with B1 cheater. After the encoded qubit is captured and decrypted, it is

teleported to B1 cheater. A N -qubit GHZ state measurement is applied on the teleported

state. The measurement results are shared among cheaters, and the encoded message

can be obtained. It is easy to see that the cheaters can produce the same correct results

21


at the same time as the prover at P .

2.6 Principle of the cheating schemes

2.6.1 Protocol A

Before discussing the reason that cheating is possible, I note that the strategy discussed in

Sec. 2.4 and 2.5 does not cheat only Protocol A that the encrypted states are BB84 states

(eigenstates of X and Z operators), but it can also cheat a more general protocol that

the encrypted states are the eigenstates of any Pauli (X , Y , or Z) operators 10. In the

two-verifier case, this new protocol can be cheated by the same procedure as described

in Sec. 2.4. There are two main reason for this: First, the teleportation byproducts,

I,X, Z, and XZ, does not change the basis of the eigenstates of Pauli operators. Second,

Bell measurement can be acted as a parity check for two states that are the eigenstates

of the same Pauli operator.

In the case of more than two verifiers, the strategy can be modified to cheat a more

general protocol that the encoded qubit is encrypted as CN . . . C2|u〉, where Ci is a Clifford

operator that transforms an eigenstate of Pauli operator to that of (possibly another)

Pauli operator. The classical information of each Ci will be sent to the prover from the

Vi verifier.

Instead of using a N -particle GHZ state, the cheaters share a 4N − 3 particles chain

cluster state 11 for conducting MBQC. B1 cheater picks the end qubit of the chain, while

other cheaters pick four consecutive qubits from the chain. As stated in Ref. [148],

arbitrary rotation of the state can be conducted by measuring three consecutive cluster

state qubits in appropriate directions, while the state can be teleported to the next qubit

in the chain by measuring the fourth qubit in the X basis. To cheat the current protocol,

each cheater conducts MBQC on the four cluster state qubits to implement Ci on the

cluster state qubit of B1 cheater. Therefore after all measurements, the qubit will be in

the same basis as the incoming qubit. Then B1 cheater measures the encoded qubit and

cluster state qubit in the Bell basis for a parity check. I note that all cluster state qubits

can be measured at the same time, because the measurements are local operations and

obviously independent to each other.

In summary, this protocol aims to encrypt the quantum message by using Clifford

operators, but it can be cheated by using MBQC. The principle for the success of the

10This more general protocol includes Protocol A because Hadamard operator is a Clifford operator.11A chain cluster state can be constructed by preparing a chain of |+〉 qubits, and then applying

Controlled-phase gate on two consecutive qubits.

22


cheating strategy is related to several properties of Clifford and Pauli operators. Firstly,

the byproducts of MBQC are Pauli operators [148]. More explicitly, the cluster state

qubit of B1 cheater, |ψout〉, after MBQC is given by [148]

|ψout〉 =N∏

i=2

(UΣiCi) |0〉 , (2.19)

where UΣiis the Pauli byproduct depending on the measurement outcomes of Bi cheater.

The next useful property is that Clifford gates are, by definition, maps a Pauli operator

to another Pauli operator, i.e.,

CP = P ′C , (2.20)

where P and P ′ are some Pauli operators. By using this properties, |ψout〉 can be ex-

pressed as

|ψout〉 = CN . . . C2UΣ|0〉 , (2.21)

where UΣ is a Pauli operator. Finally, I observe that any Pauli operator does not change

the basis on |0〉 states, but it only flips or add an unimportant global phase on the qubit,

i.e.,

|ψout〉 = eiφCN . . . C2|0 or 1〉 . (2.22)

where eiφ = ±1 or ±i. As a result, |ψout〉 will be in the same basis as the encrypted state

CN . . . C2|u〉. Because the two qubits are eigenstates of a Pauli operator, their parity

can be checked by a Bell measurement. After sharing the measurement information, all

cheaters will know the encoded message u and will report to verifiers. The whole process

takes the same or shorter time than the honest case, thus the protocol is insecure.

2.6.2 Protocol B

The security argument of Protocol B is based on the quantum no-cloning theorem [123].

However, it is not necessary to clone the state in order to remotely conduct a perfect

measurement. The problem of Protocol B is that each codeword of a GHZ state is

related to other codewords by bit-flips and phase-shifts. Since the random byproducts of

teleportation are Pauli operators, the teleported state is only bit-flipped or phase-shifted

but remaining in the codespace. As a result, a standard GHZ state measurement can

perfectly measure the teleported state.

23


2.7 Modified PBQC Protocol

In the previous section, I have discussed that the generalised Protocol A is insecure

because the encoded state is encrypted by Clifford operators. To resist the cheating

strategy discussed in Sec. 2.4 and 2.5, a natural modification to Protocol A is to encrypt

the encoded state by non-Clifford operation. One example is to transform the encoded

state as,

|0〉 → |ψ0〉 = cosθ

2|0〉+ sin

θ

2eiφ|1〉 , (2.23)

|1〉 → |ψ1〉 = sinθ

2|0〉 − cos

θ

2eiφ|1〉 , (2.24)

where the polar angles are randomly picked from 0 ≤ θ ≤ π and 0 ≤ φ ≤ 2π. The states

|ψ0〉 and |ψ1〉 are the ±1 eigenstates of n(θ, φ) · ~σ, where ~σ = Xx+ Y y+Zz is the Pauli

vector, and n(θ, φ) is a unit vector pointing along the polar angles θ and φ. I refer this

modified protocol as Protocol A′ hereafter. A similar protocol was also proposed in Ref.

[97].

While the configuration is the same as Protocol A, the only difference in Protocol A′

is that the encoded state prepared by V1 cheater is UN . . . U2|u〉, where each Ui is a single

qubit rotation. The classical information of Ui is delivered to Vi verifier, and will be

sent to the prover. It is not difficult to check that the cheating strategy demonstrated in

Section 2.4 and 2.5 cannot cheat Protocol A′ perfectly. In the two-verifier case, suppose

B1 cheater captures the state |ψi〉 at t = (d−l)/c, and teleports toB2 cheater immediately.

Although B2 cheater knows the basis from V2 verifier, the teleported state is in general

neither parallel nor anti-parallel with |ψi〉, i.e. for a non-trivial teleportation byproduct,

the matrix element

〈ψ|X(1−s2)/2Z(1−s1)/2|ψ〉 6= 0 or 1 , (2.25)

where s1 and s2 are the measurement outcomes of the teleportation measurement. There-

fore B2 cheater cannot perfectly measure the qubit before hearing the measurement out-

comes from B1 cheater, but the information of B1 cheater arrives B2 at the earliest

t = (d+ l)/c. Even if B2 cheater measures the qubit immediately at t = (d+ l)/c, correct

feedback will reach V1 no earlier than t = 2(d + l)/c, which costs more time than the

honest case. The security of Protocol A′ is hence enforced.

Similarly in the case of more verifiers, the cluster state qubit of B1 cheater will be

transformed by MBQC as

|ψout〉 =N∏

i=2

(UΣiUi) |0〉 . (2.26)

24


Since Ui’s do not belong to the Clifford group, |ψout〉 is generally not in the same basis as

|ψ0〉. Furthermore, even if the states are in the same basis, parity check by Bell measure-

ment is not possible as the states are not eigenstates of Pauli operators. Therefore, B1

cheater cannot make a perfect measurement before receiving the information from other

cheaters. Thus the total time to make the correct response is slower than the honest

case.

In practice, neither the quantum operations, quantum channel, nor measurements are

noiseless, incorrect response can be given even in the honest case. The total error rate of

a practical PBQC system ought to be lower than the rate of fail cheating, i.e. probability

that the cheaters make incorrect response. Otherwise failure of cheating may be regarded

as error caused by noise, and the PBQC protocol becomes insecure.

I now discuss the successful cheating rate of Protocol A′ under various simple cheating

schemes in the two-verifier case. First of all, I consider B1 cheater measures the encrypted

qubit in a random basis once received. Since θ and φ are random, I can assume every

measurement is made in the Z basis without loss of generality. Let us assume the mea-

surement outcome is |0〉. The probability that the encrypted state is |ψ0〉 (|ψ1〉) is equalto |〈0|ψ0〉|2 = cos2(θ/2) (|〈0|ψ1〉|2 = sin2(θ/2)). After receiving the basis information

from B2 cheater, if θ < π/2, |ψ0〉 is more likely to be the encrypted state; otherwise if

θ > π/2, |ψ1〉 is more likely. So the cheaters report |ψ0〉 when θ < π/2, and |ψ1〉 when

θ > π/2. After simple calculations, the total probability for the cheaters to make the

correct response is 75%.

Next, I consider cheaters conduct the teleportation cheating scheme in Sec. 2.4. B2

cheater measures the teleported state in the basis |ψ0〉, |ψ1〉. Consider if the result is

|ψ0〉. After knowing the teleportation measurement outcomes s1 and s2, the cheaters

announce the more probably correct result, i.e.

|v〉 =

|ψ0〉, |ψ1〉∣

∣ max(|〈ψ1|X(1−s2)/2Z(1−s1)/2|v〉|2)

. (2.27)

By averaging over θ and φ, the rate of obtaining a correct guess can be found as

1

4π

∫

[

1 + max(|〈ψ0|X|ψ0〉|2, |〈ψ0|X|ψ1〉|2) + max(|〈ψ0|Z|ψ0〉|2, |〈ψ0|Z|ψ1〉|2)

+max(|〈ψ0|XZ|ψ0〉|2, |〈ψ0|XZ|ψ1〉|2)]

dΩ . (2.28)

The numerical result of the above integral is about 85%. In general, B2 cheater can

measure in basis other than |ψ0〉, |ψ1〉. I numerically find that 85% is the highest

probability that the cheaters can get. In the case of more verifiers, MBQC requires

25


more measurements and thus produces more measurement byproducts, so the successful

cheating rate is anticipated to be lower than 85%.

The above analysis shows that Protocol A′ is secure against the entanglement attack

strategy proposed in this chapter. I note that the protocol is in fact secure against

arbitrary attack if the cheaters share only a pair of entangled qubit or qutrit. Detailed

security proof is given in Appendix A.1. Historically, our results gave the first lower

bound of entanglement resources that is required to successfully cheat a PBQC protocol:

at least two Bell pairs.

2.8 Summary

In this chapter, I have presented my work in position-based quantum cryptography, which

has been published as Ref. [108]. I have reviewed all known PBQC protocols in 2010,

and introduced the respective cheating strategies, by using which cheaters can always

produce the same response as the honest prover. The main reason for the successful

cheating is that the cheaters can share entangled resources, of which this possibility is

not considered in the proposals of PBQC protocols [43, 122]. I have further discussed

the principle and limitations of my cheating strategies. I have also proposed a modified

protocol that is secure if the cheaters share only a pair of maximally entangled qubit or

qutrit.

Nevertheless, the modified protocol was soon shown to be insecure. In fact, any PBQC

protocol can be cheated with arbitrarily high successful probability by using nonlocal

quantum measurement [172, 43, 188] or port teleportation [89, 88, 23]. In other words,

PBQC is not unconditionally secure, even if all the apparatus of the verifiers and the

honest prover is perfect.

In practice, the apparatus must be imperfect that leaves extra room for cheating. For

example, practical detectors are not 100% efficient and the quantum channel is lossy, so in

some rounds of challenge the honest prover may not receive any signal. The cheater can

make use of this property to improve their successful cheating rate. More explicitly, the

cheaters may report only in the rounds that the encrypted qubit is perfectly measured,

while they claim a loss in the rounds of measurement in a wrong basis. The verifiers

would have a high probability to receive correct response, and falsely regard the presence

of a honest prover.

In addition, the imperfection of apparatus also limits the case that PBQC is prac-

tically useful. For example, the quantum and classical operation time is finite instead

of infinitesimally short, and the quantum channel may not be vacuum that quantum

26


signals are not transmitted at the speed of light. These extra time consumptions have

to be smaller than the response time difference that verifies the prover’s location. In

other words, PBQC can verify the location of the prover within a sufficiently large area,

through which the light travelling time, l/c, is longer than the time consumption due to

the apparatus imperfection. Let us consider the refractive index of a standard optical

fiber is c′ ≈ c/1.5, so the extra time consumption is δt ≈ d/3c. Therefore the preci-

sion range of the verifiable position is at least one third of the verifier-prover separation.

When applied to verify the location of human-size object, this limitation requires a high

density of verifier station, say one per 10 m2, which is not very practical.

Because the scheme is not unconditionally secure, and it suffers from various technical

difficulties in implementation, I conclude that PBQC is not a practical quantum infor-

mation application. Nevertheless, there are still academically interesting questions that

remain open. For example, all cheating strategy may fall in a communication complexity

that cheating consumes much more resources than the honest execution of the protocol

[40]. If this claim is valid, PBQC can be conditionally secure, i.e., when the cheaters have

only bounded amount of entanglement. Besides, in the most efficient cheating strategy

known today, the amount of required entanglement scales polynomially as the degree of

freedom of the encrypted message. On the other hand, the provable minimum amount

of entanglement that is necessary for cheating scales linearly as the degree of freedom of

the encrypted message. The optimal amount of entanglement for cheating remains un-

known. Answering this question would deepen our understanding about the capability

of entanglement in communication tasks.

In addition, according to the analysis of all known cheating strategies, infinite entan-

glement is required if the message is encrypted in a state with infinite degree of freedom.

In practice, such a state is not difficult to realise; the quantum states used in continuous-

variable communication schemes [34], such as coherent states and squeezed states, have

infinite degree of freedom. It is interesting to study if the known cheating strategies can

be extended to cheat continuous-variable PBQC scheme. These studies will be related

to nonlocal measurement and port teleportation of continuous-variable quantum states,

of which the techniques is not know at this moment.

27

Chapter 3

Quantum Secret Sharing with

Continuous-Variable Cluster States

3.1 Introduction

Secret sharing is a cryptographic task aiming to distribute a secret amongst a group

of parties. It starts from a party, known as the dealer, who encodes the secret into

some information carrier according to a secret sharing protocol. A good protocol should

allow each set of parties in the access structure, which is the set of all authorised sub-

sets of parties, to faithfully reconstruct the secret; while other parties in the adversary

structure, which is the set of all unauthorized parties, are denied any information about

the secret. Classical secret sharing protocols have been proposed [31, 160] where classical

information is encoded by mathematical transformations. The protocols can be proven

to be information-theoretically secure, i.e. if the communication channels between the

dealer and the parties are secure, no information about the secret can be obtained by the

adversary structure even when they have unlimited computational power.

Following the rapid development of quantum information, the extension of secret

sharing to the quantum regime has received much theoretical attention [83, 51, 71, 170].

The objective of quantum secret sharing (QSS) is to use the quantum correlations in

well-constructed entangled states to securely transmit a set of classical or quantum in-

formation to only the access structures. As the involved parties are supposed to be

spatially well separated, an optical system is the most suitable implementation for QSS

due to its excellent mobility. Several proof-of-principle experiments have already been

demonstrated [167, 47, 102]. However, constructing a large-scale optical QSS state is

technically challenging, because the nonlinear interaction between photons is weak, and

28

Chapter 3. Quantum Secret Sharing with Continuous-Variable Cluster States

some QSS protocols require more than two quantum levels [51, 71], where the commonly

employed polarisation encoding is not applicable. Recently, Markham and Sanders [124]

proposed a unified QSS approach based on qubit cluster states [147, 148], which could

be constructed efficiently using only linear optics and post-selection [142, 37, 56].

Cluster states have another advantage that an N -mode cluster is well characterised

by N stabilisers or an N -vertex connected graph, in contrast to a general quantum state

that have to be expressed in an exponential number of superpositions. Therefore, the

theoretical construction and the security analysis of a cluster state QSS scheme could

be simplified. The idea of cluster state QSS has also been extended to odd-dimensional

states (qudits) in Ref. [95].

In this chapter, I present the work of Christian Weedbrook and myself in extending

cluster state QSS into the continuous-variable (CV) regime. While many quantum infor-

mation protocols can be optically implemented by using discrete- or continuous-variable

formalism, CV systems have the advantages that multi-partite entangled states can be

produced deterministically, and the measurement is high in fidelity using present tech-

nology. In particular, CV cluster states are proposed to be useful resources to conduct

measurement-based universal quantum computation [128, 176]. A CV cluster state can

be efficiently implemented in an optical system by various approaches including the con-

ventional method of controlled-phase (CPHASE) operation [128, 190], linear optics with

offline squeezing [173], optical parametric oscillator [130, 129, 64], and quantum nonde-

molition gate [131]. Recently, CV cluster states involving as much as 10000 optical modes

have been demonstrated experimentally [165, 189, 187]. For simplicity, I consider that

the CV cluster states are prepared by the conventional method of CPHASE operation,

though the states can be equivalently prepared by other approaches, and my result is

independent of the method of state preparation.

The main objective of this work is to investigate how CV cluster states can be used

to securely share quantum and classical secrets. Instead of directly extending the qudit

approach to the d → ∞ limit, a CV cluster state is critically different from its discrete-

variable counterparts in the sense that constructing a perfect (infinitely squeezed) CV

cluster state is practically impossible. I find that when realistic finitely squeezed cluster

states are instead utilised, QSS is still possible but the security is inevitably reduced,

i.e., the secret is not precisely recovered by the access structure while partial information

is leaked to unauthorised parties. I suggest benchmarks to evaluate the performance of

each of the QSS tasks. For the sharing of classical information, I calculate the amount

of secure key, which is used to encode the secret, that can be distilled from each cluster

state. A procedure is provided to transform the distilled state to the standard form

29


that can be analysed by the techniques in CV quantum key distribution (QKD). For

the sharing of quantum information, I estimate the number of cluster states required to

establish a high fidelity teleportation channel to transmit the secret state. The amount

of entanglement is quantified by the logarithmic negativity. In both tasks, I give two

examples to demonstrate the decoding and security analysis procedures.

As I want to focus the discussion on the application of the quantum correlations

of CV cluster states, the states received by the parties are assumed to be the same as

when prepared by the dealer, i.e., all quantum channels are ideal (noiseless and lossless).

Detections are also assumed to be perfect in fidelity.

This chapter is outlined as follows. In Sec. 3.2, I introduce QSS and classify it into

three tasks. The physical and mathematical background of CV cluster states is also

reviewed. In Sec. 3.3 and 3.4, I respectively analyse the security of classical information

sharing when the cluster state is delivered through secure and insecure channels. In Sec.

3.5, I discuss the performance of quantum state sharing. I conclude in Sec 3.6 with a

short discussion. I note that most the material in this chapter has been published in

Ref. [109].

I denote the quantities of the access structure by the subscript A, that of the adversary

structure or other unauthorised parties by E, and that of the dealer by D. I pick ~ = 1

in the following calculations, and all logarithms are to base 2.

3.2 Background

3.2.1 Quantum Secret Sharing

In the literature, the idea of QSS has been developed to serve one of the following three

tasks [124]:

CC: Classical information is shared among parties by distributing QSS states through

private (secure) channels, which are invulnerable to eavesdropping. The role of quan-

tum resources is to substitute the mathematical correlations in classical secret sharing

protocols by the quantum correlations in a QSS state.

CQ: Classical information is shared among parties by distributing QSS states through

public (insecure) channels, which are open for eavesdropping. The quantum correlation

in the QSS states is used both to detect the disturbance of eavesdropping and to share

the secret. When compared with the hybrid approach that incorporates both classical

secret sharing and QKD, the CQ scheme can reduce the cost of communication [83].

QQ: Also known as ‘quantum state sharing’, a secret quantum state is shared among

30


parties by distributing QSS states through public channels. The QQ scheme can be im-

plemented by either encoding the quantum secret into a QSS state, or by using a QSS

state to distribute entanglement between the dealer and the access structure for tele-

porting the secret state [48]. Because the former approach would allow the eavesdropper

to access the secret by capturing all the QSS states, I consider in my work the later

approach that the secret can be sent after the security of the teleportation channel is

verified.

The three tasks form a hierarchy of the required resources: a QQ state can perform

all three tasks, and a CQ state can be used for CC, while the reverse is not always true.

In principle, constructing a QQ state is versatile, but the amount of required resources

and infrastructure can be optimised according to the properties of the shared information

and the channels.

For CC and CQ, I consider that both the dealer and the access structure measure

the cluster states after state distribution. Because of the entanglement, random but

strongly correlated measurement outcomes will be obtained, from which the dealer and

the access structure can distill secure keys. The keys can then be used to encrypt the

classical secret, which will be shared through public classical channels. Therefore the

secret sharing rate, i.e., the amount of classical information securely shared in each

round of QSS, is determined by the net amount of secure key distilled from each cluster

state.

For QQ, I consider that the dealer and the access structure extract entangled states

from the cluster states. After accumulating enough extracted states, entanglement distil-

lation can be conducted to distill a more entangled state, through which the secret state

can be teleported from the dealer to the access structure with higher fidelity.

I note that in all the QSS tasks, the objective of the dealer is to securely transmit

the secret to the access structure, although the identities of the access structure are

not revealed until all QSS states have been received. Because, in a secure protocol,

the mutual information between the dealer and the access structure is larger than the

information obtained by the adversary structure, the access structure’s identities can be

authenticated using parts of the shared information. The dealer should then trust the

access structure and co-operate in subsequent post-processing of the shared QSS states.

I also note that in the limit of infinite squeezing, our cluster state scheme is not as

general as the QQ scheme proposed in Ref. [170]. However our scheme is interesting

because all three kinds of QSS are considered in a unified approach, and the resource

state is a cluster state that can be efficiently constructed and easily analysed.

31


3.2.2 Continuous-Variable Cluster States

As an analog to the discrete-variable cluster state, which is formed by preparing all qudits

in an eigenstate of the generalised Pauli X operator and then applying the CPHASE gate,

a CV cluster state is formed by first preparing all quantum modes as squeezed vacuum

states and applying CV CPHASE gates, i.e., C = expiAij qiqj. A n-mode CV cluster

state can be characterised by a n-vertices graph, where each quantum mode acts as a

vertex vi ∈ V, where V = vi, and a CPHASE operation is applied across each edge

eij ∈ E , where E = eij = vi, vj, with weight Aij [95]. The CV cluster state |Ψ〉 is

defined as

|Ψ〉 :=∏

eij∈EexpiAij qiqj|ψ0〉⊗n . (3.1)

In the infinitely squeezed case, |ψ0〉 is given by

|ψ0〉infinite = |0〉p , where p|0〉p = 0 , (3.2)

while in the finitely squeezed case,

|ψ0〉finite =√σ

π1/4

∫

e−σ2q2/2|q〉qdq , (3.3)

where |q〉q is the eigenstate of q with eigenvalue q; σ is a parameter characterising the

degree of squeezing.

3.2.2.1 Nullifier representation

Apart from the ket vector representation, an infinitely squeezed cluster state can be

characterised by its stabilisers [2, 176]. A stabiliser S of a state |ψ〉 is defined as the

operator of which |ψ〉 is an eigenstate with +1 eigenvalue, i.e., S|ψ〉 = |ψ〉. Analogous tothe discrete-variable cluster state, a n-mode infinitely squeezed CV cluster state has at

most n independent stabilisers, although any sum and product of the stabilisers is a new

stabiliser. The whole set of independent stabilisers uniquely specifies the cluster state

[74].

In a CV system, it is sometimes more convenient to study with the nullifiers than

the stabilisers. A nullifier N is defined as an operator of which |ψ〉 is an eigenstate with

eigenvalue 0, i.e., N |ψ〉 = 0. There are infinitely many choice of nullifiers because any

sum and product of nullifiers is another nullifier. For an infinitely squeezed CV cluster

32


state, I choose a set of nullifiers, which is hereafter referred as the standard set, as [74]

Ni = pi −∑

j∈NAij qj , (3.4)

where the position operators are summed over the neighbours of the vertex i in the

graph, i.e., j∣

∣(i, j) ∈ E . The standard nullifiers can be constructed by the following

procedure. Before the CPHASE operations, the squeezed vacuum modes are nullified

by pi’s. The CPHASE operation between the mode i and j transforms the nullifiers as

pi → eiAij qiqj pie−iAij qiqj = pi −Aij qj . From this procedure, it can be easily seen that all

standard nullifiers commute and are linearly independent.

3.2.2.2 Wigner function representation

As an extension to the nullifier representation, the Wigner function is a good represen-

tation of the quantum correlation of finitely squeezed CV cluster states. The Wigner

function of a single mode CV state ρ is defined as [74]

W (q, p) :=1

2π

∫ ∞

−∞exp(ipx)

⟨

q − x

2

∣

∣

∣

qρ∣

∣

∣q +

x

2

⟩

qdx , (3.5)

where the definition can be trivially generalised to multi-mode states. The Wigner func-

tion of n finitely squeezed vacuum states is given by

W0(q,p) =1

πn

n∏

i

exp(−σ2i q

2i ) exp

(

−p2iσ2i

)

, (3.6)

and that of a finitely squeezed CV cluster state is

Wc(q,p) ≡W0(q,N) =1

πn

n∏

i

exp(−σ2i q

2i ) exp

(

−N2i

σ2i

)

, (3.7)

where q = (q1, . . . qn)T , p = (p1, . . . pn)

T , and N = (N1, . . .Nn)T ; Ni is the standard

nullifier in Eq. (3.4) with the operators replaced by the respective scalar variables; the

initial degree of squeezing of each mode i is σi. In the infinite squeezing limit, i.e., σi →0 ∀i, the exp(−σ2

i q2i ) terms would converge to some constants, while the exp(−N2

i /σ2i )

terms would become Dirac delta functions, i.e.,

Winfinite(q,p) ∝n∏

i

δ(Ni) . (3.8)

33


3.2.2.3 Correlations of measurement

Let us consider an infinitely squeezed CV cluster state that is locally measured by the

operators Mi, where Mi is a linear combination of qi and pi, i.e., homodyne detection

in a rotated basis. I define the measurements are compatible to the nullifiers, if there

exists a linear combination of Mi’s that equals to a linear combination of the standard

nullifiers, i.e.,∑

i=1,n kiMi =∑

i=1,n liNi for some real ki’s and real li’s. Measurement

compatible to nullifiers is important in my studies, because∑

i=1,n kiMi is a nullifier,

and so the measurement outcomes, Mi, would be correlated as∑

i=1,n kiMi = 0. Such

correlation is originated from the entanglement in the cluster state.

Similar quantum correlations of measurements prevail in finitely squeezed CV cluster

states, but the accuracy depends on the degree of squeezing. Let us consider a finitely

squeezed CV cluster state that is measured by the same set of measurement operators

Mi, the expectation value of the measurement outcomes are statistically correlated as

in the infinitely squeezed case, i.e.,

⟨

∑

i=1,n

kiMi

⟩

=⟨

∑

i=1,n

liNi

⟩

= 0 . (3.9)

However, the variance is finite, i.e.,

⟨

∆(

∑

i=1,n

kiMi

)2⟩

=⟨

∆(

∑

i=1,n

liNi

)2⟩

=

∫

(

∑

i=1,n

liNi

)2

Wc(q,p)dnqdnp =

∑

i=1,n

l2i σ2i

2, (3.10)

but scales as σ2i that is small. The correlation (and variance) comes from the exp(−N2

i /σ2i )

terms in Eq. (3.7), which are narrow-width Gaussian functions.

In subsequent discussions, I regard a quantum correlation is “strong” if the collective

variance of the local measurement outcomes is small; otherwise the correlation regarded

as “weak”. The modes are regarded strongly correlated if their local operators produce

strong correlations. My QSS scheme is secure if the access structure is stronger corre-

lated to the secret than the unauthorised parties. According to Eq. (3.10), the local

measurement operators exhibit strong correlations if they linearly combine as a nullifier

and σi’s are small. There the secret is usually encoded in nullifiers in my scheme.

34


3.2.2.4 Cluster-class state

A class of states that shares similar properties as the CV cluster state can be constructed

by applying local Gaussian operators onto |Ψ〉. These operators linearly transform the

quadrature operators in nullifiers, as well as the quadrature parameters in the Wigner

function, as q → aq q+ bqp+ cq and p→ apq+ bpp+ cp for some real constants a, b, c that

obey the uncertainty principle. General linear transformations can be implemented by

only three kinds of basic operators [117]: displacement, squeezing, and Fourier (phase-

shift) operator.

A displacement operator D(α) shifts a nullifier by a constant factor, i.e., the com-

ponents in nullifiers are transformed as q → q +√2Re(α) and p → p +

√2Im(α). All

the displacements do not affect the measurement basis nor the variance of the quan-

tum correlations; only the expectation values of the measurement results are changed.

A squeezing operator S(γ) = exp(−ir(qp + pq)/2) scales the quadrature operators as

S†qS → γq and S†pS → p/γ, where γ = er. Linear coefficients of xi and pi in the nullifiers

will be altered that may change measurement basis in general. A Fourier operator

F (θ) = exp(−iθ(q2+p2)/2) transforms the quadrature operators as F †qF = cos θq+sin θp

and F †pF = − sin θq+cos θp. The Fourier operator changes the local measurement bases

that exhibit the quantum correlation.

I note that all CV cluster-class states are Gaussian states, which are states that the

Wigner function is a, possibly multi-variable, Gaussian function.

3.3 CC Quantum secret sharing

In the CC setting of QSS, the dealer is connected to the n parties through secure quantum

channels. A classical secret value s is encoded by displacing certain modes i of the cluster

state by some function fi(s). The value of fi(s), the strength of the CPHASE Aij, and

the neighbours of the cluster N are designed for specific access and adversary structures.

A CV cluster state can be used for CC QSS if for each access structure, there is a nullifier

containing both s and the local quadrature operators of only that access structure, i.e.,

there exists real numbers li such that

N∑

i

liNi =∑

j∈AkjMj + g(s) , (3.11)

where kj are real numbers; Mj are linear combinations of the local quadrature operators

of access structure parties; g(s) is a nontrivial function of s. On the other hand, every

35


adversary structure cannot construct a nullifier that contains both s and only their local

operators.

In the case of infinite squeezing, the access structure can obtain g(s), and thus s,

by locally measuring their modes according to Mj . The scheme is secure if the reduced

Wigner function of the adversary structure is independent of s.

In the case of finite squeezing, the access structure also measures according to Mj .

Their results are strongly correlated to s, but some information about the secret is leaked

to the adversary structure due to weak correlations. The security of the QSS scheme can

be analysed by comparing the amount of information obtained by the access structure

and the adversary structure.

The information obtained by the access structure is quantified by the mutual informa-

tion, I(D : A), between the dealer and the access structure [143]. Let the dealer chooses a

secret value s according to a probability distribution PD(s). The access structure would

not obtain exactly the same value due to the finite squeezing. The conditional probability

of obtaining a result s′ follows PA|D(s, s′). The total probability of the access structure’s

result is then given by

PA(s′) =

∫

PD(s)PA|D(s, s′)ds . (3.12)

The mutual information I(D : A) is defined as [143]

I(D : A) = H(A)−H(A|D) , (3.13)

where H(A) is the entropy of the access structure’s result, which is defined as

H(A) = −∫

PA(s′) logPA(s

′)ds′ ; (3.14)

H(A|D) is the conditional entropy of the access structure when s is known, which is

defined as [143]

H(A|D) = −∫

PD(s)PA|D(s, s′) logPA|D(s, s

′)dsds′ . (3.15)

On the other hand, the adversary structure can unite their modes through ideal

quantum channels, and can conduct any operation allowed by physics. The amount of

information leaked to the adversary structure, I(D : E), is capped by the Holevo bound

χ [143, 84], i.e.,

I(D : E) ≤ χ = S(ρE)−∫

PD(s)S(ρE|D(s))ds , (3.16)

36


where S(ρ) is the von Neumann entropy; ρE|D(s) is the state obtained by the adversary

structure if s is prepared by the dealer; ρE is the average state obtained by the adversary

structure, viz.

ρE =

∫

PD(s)ρE|D(s)ds . (3.17)

As CV cluster states are Gaussian, the reduced state of the adversary structure is

also Gaussian. The von Neumann entropy of Gaussian states can be calculated by using

their covariance matrix V , which is defined as Vij := 〈∆xi,∆xj〉/2 [176]. If the Wigner

function of an r-mode Gaussian state is known, V can be obtained through the relation

[176]

W (x) =exp(−1/2(x− x)TV −1(x− x))

(2π)r√detV

, (3.18)

where x = (q1, p1, . . . , qr, pr)T ; x =

∫

xW (x)d2rx. Covariance matrices can be charac-

terised by their symplectic spectrum νk, which is equal to the eigenspectrum of the

matrix |iΩV | [176], where

Ωi,j =

1 if i = 2k − 1, j = 2k ,

−1 if i = 2k, j = 2k − 1 ,

0 else,

(3.19)

for k = 1, . . . , r. The von Neumann entropy is calculated by

S(ρ) =r∑

i

g(νk) , (3.20)

where g(ν) :=

(

ν +1

2

)

log

(

ν +1

2

)

−(

ν − 1

2

)

log

(

ν − 1

2

)

. (3.21)

In the case that the covariance matrices of ρE and ρE|D are independent of s, their

respective von Neumann entropies are also so. Then the Holevo bound can be simplified

as,

I(D : E) ≤ S(ρE)− S(ρE|D) . (3.22)

The minimum secret sharing rate in each round of the protocol is thus

Kcc = I(D : A)− I(D : E) . (3.23)

After m rounds of state distribution, mKcc secret keys can be distilled from the strongly

correlated random numbers s and s′ for sharing the classical secret [20].

As examples, I demonstrate in Appendix A.2 the procedure for calculating Kcc of

37


the (2,3)- and the (3,5)-protocol. I note here a few interesting findings regarding the

examples. Firstly, the secret sharing rate is positive when the squeezing parameter σ is

smaller than some threshold. The threshold for both protocols we considered are around

σ ≈ 1. The results show that a secure CC QSS can be implemented with only weakly

squeezed resources, hence the requirement of infinitely squeezed states can be relaxed.

Besides, on the contrary to common beliefs that a CV state with σ = 1 cannot

transmit secure information, the CC secret sharing rate is non-zero in some scenarios

even when σ ≥ 1. The result is not surprising in cluster state QSS, because implementing

a CPHASE requires the initial modes to be squeezed [173]. In fact, a two-mode cluster

state can be easily shown to be local-unitarily equivalent to a finitely-squeezed Einstein-

Podolsky-Rosen (EPR) state unless σ → ∞ [34].

In addition, although any access structure collaboration can obtain s in the infinite

squeezing case, surprisingly different collaborations obtain different secret sharing rate

in the finite squeezing case. This heterogeneity is related to the entanglement structure

of the cluster state. In practice, the dealer has to consider the disadvantage of certain

collaborations when applying CC QSS.

3.4 CQ Quantum secret sharing

In the CQ setting of QSS, the dealer is connected to the parties through insecure quantum

channels, so that unauthorised parities, which include adversary structure and eavesdrop-

per who does not involve in the protocol, can manipulate all the modes sent from the

dealer. The CC protocol mentioned in Sec. 3.3 is insecure in this setting, because the

unauthorised parties can capture and measure the modes to obtain s. This eavesdropping

can be intractable if the adversary structure resends to the access structure an infinitely

squeezed state with the same s encoded, so that the access structure have the same

measurement results as the adversary structure.

Here I modify the CC protocol for the CQ setting. I first present an entanglement-

based protocol, and discuss how it can be reduced to a mixed-state protocol that reduces

the resource requirement. Instead of constructing a n-mode cluster state and encoding a

classical secret s into the state, the dealer prepares an (n+1)-mode standard cluster state

(cluster states with standard nullifiers in the infinite squeezing case), where n of the modes

are delivered to the parties while the dealer keeps the remaining one, denoted as mode

D. A good CQ protocol should produce a much stronger quantum correlation between

the dealer and the access structure than that between the dealer and the unauthorised

parties.

38


Here I make two assumptions to simplify the the security analysis, but these assump-

tions will be relaxed at the end of this section without compromising the secret sharing

rate. Firstly, I assume the access structure parties are connected by secure and ideal

quantum channels, so the modes can be sent to one party, say party h, with perfect

fidelity. I also assume both the dealer and the access structure have quantum memories,

so the cluster states delivered in each round are stored with perfect fidelity for subsequent

quantum operations and measurements.

After each round of state distribution, a strongly correlated entangled state is shared

between the dealer and the access structure. Let us consider the strong correlation is

represented by the two nullifiers, pD − QA and qD − PA, which are linear combinations of

the standard nullifiers. QA and PA are linear combinations of only the access structure

parties’ local q and p operators. By applying a global operation, UA, on all the modes

at party h, QA and PA are transformed to single mode operators qh and ph. As a result,

the strong correlations with mode D are transferred to mode h.

After all the rounds of cluster state distribution, the stage of parameter-estimation

ensues. The dealer or party h randomly selects half of the shared modes for measurement,

and the selection is announced. Both the dealer and party h measures the selected modes

in either the x and p basis. The measurement outcomes are announced for characterising

the unmeasured states.

In the infinitely squeezed case, the estimated parameters should indicate that the

state between mode D and mode h is maximally entangled 1. The dealer and party

h measure each residual modes randomly in either the q or p basis, the basis is then

announced. Each measurement outcome is a random number on the real axis, and the

outcomes are the same if the measurement bases are matching, i.e., one party measures

in q while the other measures in p. The common random numbers can be used as secure

keys to encode the secrets. Because the state was maximally entangled, no information

is leaked to unauthorised parties.

In the finitely squeezed case, although modeD and mode h are strongly correlated, the

state of adversary structure is still weakly correlated with mode D. As to be discussed,

local quantum operation is applied on each residual mode to rectify the covariance of the

state according to the estimated parameters. The dealer and party h then measure each

mode in either the x or p basis, and announce the basis. Unlike the infinitely squeezed

case, post-processing is required to extract secure keys from the correlated measurement

outcomes due to two reasons. First, even if the measurement basis is matching, the mea-

1Here I refer to ‘maximally entangled’ as an entangled state originating from two infinitely squeezedmodes.

39


surement outcomes of the dealer and the access structure are merely strongly correlated

but not exactly equal. Besides, partial information about the outcomes is leaked to the

unauthorised parties due to the non-uniform distribution of the measurement outcomes,

and the weak entanglement between the unauthorised parties’ and the dealer’s modes.

In the following, I employ the security analysis techniques from CV QKD [3] to

estimate the minimal amount of secure key that can be distilled from each cluster state,

which is equivalent to the secret sharing rate in each round of CQ QSS.

3.4.1 Equivalence of CQ Quantum Secret Sharing and QKD

I now show why CQ QSS and CV QKD can be analysed by using the same techniques.

Let us consider that before the CPHASE operation, mode D is squeezed with σD while

all other modes are squeezed with σ. Suppose mode D is connected to N neighbours

after the cluster state formation, the reduced Wigner function of mode D is

WD(qD, pD) =σσDe

−σ2Dq2D

π√

N + σ2σ2D

exp(

− p2Dσ2D +N/σ2

)

. (3.24)

WD is the same as the reduced Wigner function of a two-mode cluster state |CN〉, wheremode D is connected to a mode u that is squeezed with σ/

√N . Because both the CQ

cluster state and |CN〉 are pure, the amount of entanglement between mode D and the

cluster state modes to be delivered is the same as the amount between the two modes in

|CN〉.As in common security analysis of QKD, I grant the unauthorised parties the full

power to manipulate the modes sent from the dealer. Then there will be no difference

for the dealer to prepare the CQ cluster state or |CN〉, because the unauthorised parties

can transform the delivered cluster state modes to mode u or vice versa. Then the CQ

state delivery is equivalent to the following scenario: The dealer first prepares |CN〉 anddelivers mode u through an insecure quantum channel. The unauthorised parties capture

mode u, entangle it with ancillae, and forward some modes to the access structure.

The access structure’s modes are then gathered at party h. After the operation UA,

modes other than mode h are still weakly correlated with mode D. For simplicity, these

weak correlations are neglected in our analysis, i.e., all modes except h are traced out.

This action only reduces the quantum correlation between the dealer and the access

structure, thus the security is not unphysically improved. Now the CQ protocol is effec-

tively reduced to a CV QKD protocol: The dealer first prepares a two-mode Gaussian

state, |CN 〉, and delivers one mode. The access structure finally gets a mode h that

40


D

D u

D

h

D h

Decoding

Delivered mode

captured by adversary

Tracing out

modes besides h

D

Deliver strong

correlation to mode h

Local unitary

equivalent

Modes delivery

D u

QSSQKD

Dealer

Unauthorized parties

Authorized parties

Figure 3.1: Strategy for computing the secret sharing rate using CV QKD techniques.Strongly (weakly) correlated modes are linked by solid (dotted) lines. The procedure ofCQ QSS is shown on the right while that of QKD is shown on the left. The key idea isthat both QKD and QSS have the same initial (pure entangled state with parts delivered)and final resources (two strongly correlated modes between the dealer and the authorisedparties.)

remains strongly correlated with mode D, but the quantum correlation is reduced due

to the entanglement with the environment controlled by the unauthorised parties. The

degradation of quantum correlations in the encoding and the decoding processes in CQ

QSS can be analogous to the loss and noise when transmitting an EPR state through an

imperfect channel in QKD. The whole idea is summarised schematically in Fig. 3.1.

3.4.2 Secret Sharing Rate

In the unified picture of CV QKD, a finitely squeezed two-mode squeezed state is prepared

by the dealer and delivered to the authorised party through an imperfect channel [68, 176].

41


Both parties measure some of the delivered states to estimate the covariance matrix,

V , of the unmeasured states. By using the fact that Gaussian states minimise the

distillable secure key for every state having the same V [184, 68], the assumption that the

unmeasured states are Gaussian upper-bounds the information leakage to unauthorised

parties. Because a Gaussian state is completely characterised by its covariance matrix,

the secure key rate can be deduced from only V . For realistic channels that are usually

symmetrical for quadratures x and p, V can be expressed in a standardised form as

V =

(

V I cZ

cZ V ′I

)

, (3.25)

where I and Z are the 2×2 identity and Pauli Z matrices respectively; V is the variance

of the undelivered mode of the dealer; V ′ is the variance of the mode received by the

authorised party; c accounts for the correlation between the two modes. V can be

characterised by only V and two channel parameters: the transmittance, τ , and the

noise, χ, which are defined by the relations

c =

√

τ(V 2 − 1

4) ; V ′ = τ(V + χ) . (3.26)

To calculate the minimum secret sharing rate of the CQ QSS, in the parameter-

estimation stage the dealer and the access structure construct the covariance matrix by

measuring some of their modes. In order to employ the analysis of CV QKD, subsequent

local operations are applied on the unmeasured modes to transform their covariance

matrix to the standard form. This can be done in three steps: Firstly, any covariance

matrix can be transformed by local unitaries as [54] 2

VD 0 0 c1

0 VD c2 0

0 c2 VA 0

c1 0 0 VA

. (3.27)

Next, the access structure squeezes the modes to balance the off diagonal terms, i.e., the

2This can be done by applying FA(π/2) onto the standard form I in Ref. [54]. I consider both c1 andc2 are positive, which is the case that strong correlation is retained in the cluster state.

42


covariance matrix becomes

VD 0 0√c1c2

0 VD√c1c2 0

0√c1c2 Vq 0

√c1c2 0 0 Vp

. (3.28)

Finally, the variances of qA and pA can be balanced by a ‘state-averaging’ process 3 . Let

us consider that the dealer randomly divides the unmeasured states into two sets, and

the choice of division is announced. In one set, the dealer applies a Fourier operator,

FD(−π/2), on each mode that transforms the quadrature operators as qD → pD and

pD → −qD. In the other set, the access structure applies FA(−π/2) on each mode that

generates the transformation qA → pA and pA → −qA. Subsequently, the choice of

division is discarded. The state will be transformed as

ρDA → 1

2FD(−π/2)ρDAF

†D(−π/2) +

1

2FA(−π/2)ρDAF

†A(−π/2) , (3.29)

and the covariance matrix becomes

VD 0√c1c2 0

0 VD 0 −√c1c2√

c1c2 0 (Vq + Vp)/2 0

0 −√c1c2 0 (Vp + Vq)/2

, (3.30)

which is in the standard form. The analogous τ and χ can then be obtained according to

Eq. (3.25). For pedagogic purposes, I demonstrate in Sec. A.3 the procedure of getting

the standardised covariance matrix for different collaborations in the (2,3)- and (3,5)-CQ

protocols.

I assume the QSS protocol is direct reconciliation, i.e., the measurement result of

the dealer is the secret value that has to be estimated by the access structure, but the

secret sharing rate of a reverse reconciliation protocol can be easily calculated by similar

procedure [3]. The secret sharing rate, KCQ, is given by the secure key rate in the

analogous CV QKD protocol as [3]

KCQ = I(D : A)− I(D : E) , (3.31)

3In spite of the simplicity, this action would sacrifice some quantum correlations, and hence reducethe secret sharing rate. As my aim is to demonstrate the possibility, but not the optimality, of performingCQ QSS with CV cluster states, the employment of state-averagings is appropriate within the scope ofthe current work.

43


where I(D : A) is the mutual information between the dealer and access structure; the

information obtained by the unauthorised parties is given by I(D : E), which is capped

by the Holevo bound.

The mutual information I(D : A) can be calculated by comparing the variance of

mode h with and without knowing the measurement results of mode D. In terms of the

analogous channel parameters, the mutual information is given by [3]

I(D : A) =1

2log( V + χ

χ+ 14V

)

. (3.32)

In direct reconciliation protocols, the Holevo bound of the unauthorised parities’ infor-

mation is defined as

I(D : E) = S(E)− S(E|D) , (3.33)

where S(E) is the von Neumann entropy of the unauthorised parties’ state; S(E|D) is the

conditional von Neumann entropy if the measurement result of the dealer is known. As

the unauthorised parties can control the environment that purifies the whole system, the

entropy of the unauthorised parties is the same as that of the system DA, i.e., S(E) =

S(DA). The entropy can be calculated by using Eq. (3.20), i.e., S(DA) = g(ν+) + g(ν−)

[176], where the symplectic spectrum of V , ν+, ν−, is given by

ν± =1

2

(

√

(V + V ′)2 − 4c2 ± (V − V ′))

. (3.34)

Similarly, because the state of system AE is pure after system D is measured, the

conditional entropy S(E|D) is the same as S(A|D). The covariance matrix of system A

after the measurement of the dealer is given by [59, 62]

VA|D =

(

V − c2/V 0

0 V ′

)

, (3.35)

where the symplectic eigenvalue is

νc =

√

V ′(

V ′ − c2

V

)

= τ

√

(V + χ)

(

1

V+ χ

)

. (3.36)

Hence we get S(E|D) = g(νc).

After m rounds of state distribution, the dealer and the access structure can distill

mKCQ secret keys from the strongly correlated measurement outcomes for sharing the

classical secret [20].

44


In the (2,3)- and (3,5)-protocols I have analysed, secure keys can be distilled if the

squeezing parameter is smaller than a threshold limit 4. The threshold values are about

σ ≈ 1 in the examples I considered. Just as in the CC case, our results show that

nonzero secret sharing rate can be obtained with finitely squeezed resources. Disparity

of the secure key rate in different collaborations is also observed in CQ protocols, which

is also due to the structure of the entangled state. Besides, the secret sharing rate is non-

zero in some cases even when σ ≥ 1. As I have discussed in the CC case, a two-mode

cluster state is still entangled even if the initial state is not squeezed. The entanglement

between the dealer and party h imposes strong quantum correlations from which secure

keys can be extracted.

3.4.3 Simplification

The above security analysis of CQ protocols is studied under three assumptions: (i) the

resource is a (n + 1)-mode cluster state; (ii) quantum memory is available to store the

delivered modes until all states are received; (iii) ideal quantum channels are available

between the access structure parties. Here I show that these assumptions can be relaxed

without compromising the security.

3.4.3.1 Mixed State Approach

Because the distributed modes are unaffected by any local operation of the dealer, the

state obtained by the parties is independent of whether mode D has been measured.

Therefore, instead of preparing a (n+ 1)-mode cluster state |Ψ〉 and measuring mode D

afterwards, the dealer can simulate the consequence of the measurement by distributing

a n-mode state that is the same as the measured |Ψ〉. More explicitly, let us consider

if the dealer intends to measure in the q basis, then he can instead prepare the pure

state (〈s|qD)|Ψ〉, where |s〉qD is a qD eigenstate with the eigenvalue s. Similarly in the p

measurement rounds, the dealer can prepare (〈s|pD)|Ψ〉, where |s〉pD is the pD eigenstate

with the eigenvalue s. Other parties cannot distinguish the mixed state from |Ψ〉 if s is

picked according to the probability distributions

PqD(s) =

∫

WD(s, pD)dpD =σD√πe−σ2

Ds2 ; (3.37)

PpD(s) =

∫

WD(qD, s)dqD =e− s2

σ2D

+N/σ2

√π√

σ2D +N/σ2

. (3.38)

4Except in the 2, 3 collaboration in (2,3)-protocol that secure keys can be distilled for any squeezingparameter, because the entanglement of the adversary structure is completely removed.

45


These are the probability distribution of dealer’s measurement outcomes on |Ψ〉, whichare obtained by integrating the Wigner function in Eq. (3.24).

In the simulated q measurement rounds, the infinitely squeezed (〈s|qD)|Ψ〉 is charac-terised by the nullifiers

N qi = pi −AiDs−

∑

j∈NAij qj , (3.39)

where i = 1, . . . , n; AiD = 0 if the mode i is not a neighbour of mode D. The nullifiers

N qi is the same as Ni except the operator qD is replaced by the simulated measurement

outcome s. In the finitely squeezed case, the state can be characterised by the Wigner

function WqD(q1, p1, . . . , qn, pn), which is obtained by tracing out pD and replacing all qD

by s in the Wigner function of |Ψ〉. Because WqD is the same as Wc in Eq. (3.7) with

the nullifiers N qi , (〈s|qD)|Ψ〉 can be constructed by displacing a finitely squeezed cluster

state.

In the simulated p measurement rounds, the infinitely squeezed (〈s|pD)|Ψ〉 is charac-terised by the nullifiers

Np1 =

∑

j∈NAjDqj − s ; Np

i = Ni − Ni−1 , (3.40)

where j = 2, . . . , n. In the finitely squeezed case, the state can be characterised by the

Wigner functionWpD(q1, p1, . . . , qn, pn), which is obtained by tracing out qD and replacing

all pD by s in the Wigner function of |Ψ〉. However, WpD cannot be represented by Wc

with the nullifiers Np1 , therefore (〈s|pD)|Ψ〉 is generally not a cluster state. Nevertheless,

(〈s|pD)|Ψ〉 is a Gaussian state that can be efficiently prepared by squeezed vacuum states,

displacement operators, and linear optical elements [173].

3.4.3.2 Classical Memory

When estimating the secret sharing rate, I require the delivered modes to be rectified so

that the covariance matrix is in the standard form. The modes are stored in quantum

memories until the covariance matrix is constructed from parameter-estimation. Here I

show that the measurement probability distribution of the transformed state ρ′′ can be

obtained by: first measuring the original state ρ, and then subjecting the measurement

results to classical manipulations. Therefore the delivered modes can be measured before

the parameter-estimation stage, quantum memory is thus not necessary.

The rectifying process involves two stages: local-squeezing and state-averaging. After

state-averagings, ρ′′ becomes a mixture of FD(−π/2)ρ′F †D(−π/2) and Fh(−π/2)ρ′F †

h(−π/2).By definition, the Wigner function of ρ′′, W ′′, can be written as the sum of the Wigner

46


function of ρ′, W ′, as

W ′′(qD, pD, qh, ph) =1

2W ′(pD,−qD, qh, ph) +

1

2W ′(qD, pD, ph,−qh) . (3.41)

Let us consider that the dealer measures ρ′′ in qD and party h measures in qh, the

probability of obtaining measurement outcomes y1 and y2, P ′′qD,qh

(y1, y2), is given by

P ′′qD ,qh

(y1, y2) =

∫

W ′′(y1, pD, y2, ph)dpDdph

=1

2

∫

W ′(pD,−y1, y2, ph)dpDdph +1

2

∫

W ′(y1, pD, ph,−y2)dpDdph

=1

2P ′

pD,qh(−y1, y2) +

1

2P ′

qD ,ph(y1,−y2) , (3.42)

where the last equality involves renaming of variables; P ′xD,xh

is the joint xD, xh mea-

surement probability of ρ′. Similarly, the probability of another strongly correlated mea-

surement, pD, ph, can be expressed as

P ′′pD,ph

(y1, y2) =1

2P ′

qD,ph(y1, y2) +

1

2P ′

pD ,qh(y1, y2) . (3.43)

These two relations indicate that the measurement probability distributions after

state-averagings are not different from mixing some measurement probability distribu-

tions before state-averagings. Let us consider that the dealer and party h randomly

measure ρ′ in x and p basis. Half of the qD, ph outcomes and half of the pD, qhoutcomes are picked to mimic the qD, qh measurement of ρ′′. For the qD, ph half, all

the ph outcomes are multiplied by −1 and then regarded as qh outcomes; for the pD, qhhalf, all the pD outcomes are multiplied by −1 and then regarded as qD outcomes. After

combining these two sets of data, the probability Pm of getting y1, y2 is given by

Pm(y1, y2) =1

2P ′

pD,qh(−y1, y2) +

1

2P ′

qD ,ph(y1,−y2) , (3.44)

which is the same as Eq. (3.42). The pD, ph measurement probability of ρ′′ can be

mimicked by similar procedures.

In the local-squeezing stage, the dealer and party h apply local squeezing operations,

SD(γD) and Sh(γh) , to balance the variance of mode D and the coherent terms. The

Wigner function of the state ρ is transformed as

W (qD, pD, qh, ph) →W ′(qD, pD, qh, ph) =W

(

qDγD, γDpD,

qhγh, γhph

)

. (3.45)

47


Let us consider that ρ′ is measured in qD and ph basis, the probability of obtaining

the outcomes y1, y2 is changed to

PqD,ph(y1, y2) → P ′qD,ph

(y1, y2) =

∫

W

(

y1γD, γDpD,

qhγh, γhy2

)

dpDdqh

=γhγD

PqD,ph

(

y1γD, γhy2

)

, (3.46)

where P is the probability distribution when measuring ρ. Similarly, the probability

distribution of pD and qh measurement is changed to

PpD,qh(y1, y2) →γDγh

PpD ,qh(γDy1,y2γh

) . (3.47)

Measurement results of qD, qh and pD, ph are sifted as they are merely weakly corre-

lated.

In fact, physically squeezing the state is not necessary because the transformations in

Eqs. (3.46) and (3.47) can be conducted by classically scaling the measurement outcomes.

Consider that every qi measurement outcome is scaled by 1/γi, and every pi measurement

outcome is scaled by γi. The old probability, P, of a q measurement outcome lying in the

range [y, y+ dy], is equal to the new probability, Ps, of a scaled outcome in the range of

[γy, γy + γdy]. Thus we have the relation P(y)dy = P ′(γy)γdy for q measurement, and

similarly P(y)dy = P ′(y/γ)dy/γ for p measurement. By eliminating the common factors

and redefining variables, we get

PsqD,ph

(y1, y2) =γhγD

PqD,ph

(

y1γD, γhy2

)

and PspD,qh

(y1, y2) =γDγh

PpD,qh

(

γDy1,y2γh

)

.

(3.48)

The above probability distributions are the same as Eq. (3.46) and (3.47).

3.4.3.3 Local Measurement

I have assumed the access structure parties have forwarded their modes to a single party

for global operations. Here I show that the measurement results of the dealer and the

access structure remains strongly correlated even if the access structure conducts local

measurements only.

Let us recall that the strong correlation is represented by the nullifiers pD − QA and

qD − PA. Because they are linear combinations of standard nullifiers, both the operators

48


QA and PA are sums of local operators, i.e.,

QA =

n∑

j

kqjMqj ; PA =

n∑

j

kpjMpj , (3.49)

where kqj and kpj are real coefficients; M qj and Mp

j are rotated quadrature operators

of mode j. After agreeing the measurement basis to be QA or PA, the access structure

parties homodyne detect their modes according to the basis M qj or Mp

j . The measurement

results are then shared among access structure through secure classical channels.

Without loss of generality, I consider that the access structure has chosen to mea-

sure QA. The resultant outcome QA is a linear combination of the local measurement

outcomes M qj , i.e., QA =

∑nj k

qjM

qj . The strong correlation is observed from the joint

probability distribution of pD and QA, which will be shown the same as the joint prob-

ability distribution of pD and qh. Let us consider that the Wigner function of the state

of the dealer and the access structure, WDA(qD, pD, qA,pA) where qA and pA are the

quadrature variables of the access structure, is obtained by tracing out the unauthorised

parties’ contributions in Wc in Eq. (3.7). When rewritten in terms of the new variables

M q = M qj and ∗M q = ∗M q

j , the Wigner function becomes

WDA(qD, pD, qA,pA) ≡W ′DA(qD, pD,M

q,∗M q) , (3.50)

where ∗M qj is the complementary variable ofM q

j , i.e., the corresponding operators satisfy

[M qj , ∗M q

j ] = i. The choice of ∗M qj is not unique, but we can pick the set that PA can

be written as a linear combination of ∗M qj .

I construct another set of variablesQ = QA, Q2, . . . , Qm and P = PA, P2, . . . , Pm,where Q (P ) involves linear combinations of M q

j (∗M qj ) only; and the corresponding

operators obey the commutation relations: [Qj , Pl] = iδjl, [Qj , Ql] = 0, and [Pj , Pl] =

0. Such a construction of variables is possible as there exists unitary operators that

transform M q to Q and ˆ∗Mqto P while preserving the commutation relations. In

terms of Q and P , the Wigner function can be rewritten again as

W ′DA(qD, pD,M

q,∗M q) ≡W ′′DA(qD, pD,Q,P ) . (3.51)

The local measurement outcomes follow a classical probability distribution P ′′DA,

which is obtained by tracing out the complementary components, i.e.,

P ′′DA(pD,M

q) =

∫

W ′DAdqDd

m(∗M q1 ) =

∫

W ′′DAdqDd

mP , (3.52)

49


where the last equality is imposed because P is a linear combination of ∗M q1 only . The

probability distribution of QA is obtained by tracing out the other independent variables

in P ′′DA, i.e.,

PDA(pD, QA) =

∫

P ′′DAdQ2dQ3 . . . . (3.53)

On the other hand, let us consider that the access structure parties’ modes are trans-

ferred to party h. The strong quantum correlation is transferred to mode h by applying

the decoding sequence, i.e., a global operation UA, which transforms QA → U †AQAUA = qh

and PA → U †APAUA = ph. Other operators are transformed as Qj → yj and Pj → zj .

Using the definition in Eqs. (3.50) and (3.51), the Wigner function becomes

WDA(qD, pD, q,p) →W ′′DA(qD, pD, qh, ph,y, z) , (3.54)

where y = (y2, . . . , ym)T and z = (z2, . . . , zm)

T . Because Qj and Pj commute with both

QA and PA, the transformed operators yj and zj do not contain any attributes of mode

h. The joint probability distribution of pD and qh is obtained from the Wigner function

after tracing out the modes other than mode D and h, as well as the complementary

variables qD and ph, i.e.,

PDA(pD, qh) =

∫

W ′′DAdqDdphd

m−1ydm−1z . (3.55)

The probability distribution in Eqs. (3.53) and (3.55) are deduced by different proce-

dures. The former one is deduced by first obtaining the classical probability distribution

of all local measurements, and then extracting the probability distribution of the classical

variable QA; the later one is deduced by first achieving the Wigner function of the trans-

formed quantum state, and then obtaining the measurement probability of the operator

qh. However, Eqs. (3.53) and (3.55) are mathematically equivalent because their overall

derivations are the same: tracing out all quadrature variables in the Wigner function

except those specifying the strong correlation. Similar analysis can be applied to the

correlation between qD and PA. As a result, the access structure can obtain the same

covariance matrix as I have previously discussed. Hence the secret sharing rate remains

unchanged even if the access structure’s modes are measured locally.

3.4.3.4 Simplified CQ Protocol

By incorporating the above ideas, the CQ protocol can be simplified as follow: a (n+ 1)

cluster state or a n mode mixed state is prepared by the dealer and delivered. Parties

50


in the access structure have agreed on the measurement basis in each round, local mea-

surements are conducted on each received modes. The classical measurement results are

shared among the access structure through secure classical channels. Both the dealer

and the access structure announce half of the results to estimate the covariance matrix;

while the other half is scaled and mixed so the covariance matrix is in the standard form.

The variance and the analogous channel parameters are then recognised for calculating

the secret sharing rate. Finally secure keys are distilled from the strongly correlated

measurement outcomes, the key are then used for sharing classical secrets.

I end this section with two comments. Firstly, although the quantum channels for

delivering cluster states are assumed to be ideal, I believe a modified version of my

protocol would allow secure CQ QSS with realistic (lossy and noisy) channels. The

covariance matrix of the delivered modes can still be obtained by parameter-estimation,

and subsequent classical manipulations can always scale the measurement results to obey

the standard covariance matrix.

Secondly, in all the examples I have studied, the entanglement with the unauthorised

parties’ modes only add noise to the access structure parties’ modes, while the anal-

ogous transmittance remain 1. This result is surprising in the scenario of CV QKD,

because imperfection is always simulated by adding noise into a beam splitter which

reduces transmittance. I believe this phenomenon originates from the distinctive entan-

glement structure in the resource states: a cluster state is employed in CQ QSS, while

the information carrier in CV QKD is an EPR state.

3.5 QQ Quantum secret sharing

In the QQ setting of QSS, the dealer shares a secret quantum state among parties by

delivering a multipartite entangled state. The channels connecting the dealer and the

parties can be insecure, so the unauthorised parties can manipulate all the delivered

states. In an ideal QQ protocol, the access structure can recover the secret state with

perfect fidelity, while the unauthorised parties cannot get any information about the state

due to the quantum no-cloning theorem [158].

My QQ QSS scheme is a generalisation of the CQ protocol. The dealer prepares a

(n + 1) mode cluster state, of which n modes are distributed to the parties while one is

kept by the dealer. After forming the collaborations, the access structure parties forward

their modes to party h. I assume the parties are connected by secure quantum channels,

so the access structure can combine their modes without being eavesdropped. A global

operation is then applied to extract a strongly entangled state between mode D and a

51


single mode h.

For an infinitely squeezed QQ cluster state, the strong correlation between the dealer

and the access structure is represented by the nullifiers pD − QA and qD − PA. After

all modes are gathered in party h, a decoding operation UA is applied to transfer the

quantum correlation to mode h, i.e., QA → U †AQAUA = qh and PA → U †

APAUA = ph. I

note that both the nullifiers, QA and PA, and the operation UA are the same as that in

the corresponding CQ protocol in Sec. 3.4.

The transformed nullifiers, pD−qh and qD−ph, indicate that the dealer and party h are

sharing an infinitely squeezed two-mode cluster state, which is a CV maximally entangled

state. By jointly measuring the secret state and the two-mode cluster, the dealer can

teleport the secret state to mode h. After appropriate error correction according to the

dealer’s measurement results, party h can revert the secret state with perfect fidelity.

In the finite squeezing case, party h conducts the same UA to transform the strong

correlation to mode h. However, mode D and mode h are not maximally entangled,

because their states are finitely squeezed and are weakly entangled to other modes. Con-

ducting teleportation using the non-maximal entanglement will reduce the fidelity of the

teleported state. The inaccurately shared secret state may indicate a reduction of secu-

rity of the QQ QSS, because some information about the secret state would be leaked

through the measurement results announced by the dealer, and through the states held

by the adversary structure that are weakly entangled with the teleported state.

Instead of conducting teleportation after each round of QQ QSS, I consider the ex-

tracted state is stored in quantum memories. After several rounds of the QQ protocol, a

more entangled state can be distilled from the stored extracted states through CV entan-

glement distillation [63, 38, 58, 132, 4]. Although distilling a maximally entangled CV

state is impossible due to the infinite required energy, the enrichment of entanglement

can enhance the fidelity of the teleportation.

The amount of entanglement of the distilled state is determined by that of each

extracted state, as well as the number of extracted states accumulated in the quantum

memory. We quantify the amount of entanglement by the logarithmic negativity E [174],

which is the upper bound of the distillable entanglement. The logarithmic negativity of

a state ρ is defined as

E(ρ) = log ||ρTA ||1 , (3.56)

where the superscript TA denotes a partial transpose of the density matrix; || · ||1 is

the trace norm. If ρ is a two-mode Gaussian state with a covariance matrix V , the

52


logarithmic negativity can be calculated as

E(ρ) =∑

k

F (νk) , (3.57)

where F (x) = − log(2x) if x < 1/2, and F (x) = 0 if x ≥ 1/2; νk is the symplectic

spectrum of V , which is defined as [34]

V =

(

I 0

0 Z

)

· V ·(

I 0

0 Z

)

. (3.58)

The covariance matrix V can be obtained by randomly measuring some of the stored

states in either q or p.

Because logarithmic negativity is additive [174], at least E0/E extracted states with

logarithmic negativity E is required to distill a two-mode squeezed vacuum state with

logarithmic negativity E0. As examples, I demonstrate in Appendix A.4 the procedure

of extraction, and the calculation of extracted entanglement in each round of the (2,3)-

and the (3,5)-protocols. Just as in the CC and CQ cases, finite entanglement can be

extracted when the squeezing operator is smaller than a threshold, and the amount of

extracted entanglement is different for different collaborations.

I note that logarithmic negativity is additive but not strongly superadditive [184],

so the amount of entanglement may be overestimated if the access structure’s modes in

different rounds are entangled [146], i.e., when the unauthorised parties conduct coher-

ent attacks on the delivered modes. In that case, the amount of entanglement should be

characterised by other strongly superadditive entanglement measures, such as distillable

entanglement and squashed entanglement [184]. However, logarithmic negativity is ap-

plicable in the current case because the quantum channels are assumed to be ideal, i.e.,

the access structure is expected to get the same states as prepared by the dealer, which

are individually prepared in each round. The adversary structure parties get only the

information about the shared secret through their modes obtained in each round, which

is effectively a collective attack.

3.6 Conclusion

In this work, I extended the unified cluster state quantum secret sharing framework

proposed in [124, 95] into the continuous-variable regime. I proposed that all three tasks

of quantum secret sharing can be implemented by CV cluster states. Although a QQ

53


protocol can be used to conduct CC and CQ, simplifications in the later two scenarios

can reduce the requirement of resources. For a CC protocol involving n parties, only

n-mode cluster states are needed, and the states can be measured once received. A CQ

protocol requires either a mixture of two n-mode Gaussian states or a (n + 1)-mode

cluster state. The states can be locally measured once it is received. A QQ protocol

requires (n+ 1)-mode cluster states. The states have to be transferred to one party and

accumulated in quantum memories for entanglement distillation.

On the contrary to discrete-variable systems, where no known physical principle hin-

ders the creation of a maximally entangled state, the creation of a maximally entangled

CV state requires infinite energy, and is thus not practical. Finitely squeezed states are

realistic substitutes for the maximally entangled resources, but the non-maximal entan-

glement would leak information about the shared secret to the unauthorised parties. I

proposed computable measures to account for the security of each of the three tasks

of quantum secret sharing. The secret sharing rate of a CC protocol is the difference

between the mutual information between the dealer and the access structure, and the

adversary structure’s information that is capped by the Holevo bound. The secret shar-

ing rate of a CQ protocol can be computed by calculating the secure key rate of the

analogous QKD protocol. The performance of a QQ protocol can be determined by the

amount of extracted entanglement between the dealer and the access structure.

Although I have demonstrated the analysis of only the (2,3)- and the (3,5)- proto-

cols that are both threshold protocols [51], the technique is applicable to non-threshold

protocols because the security analysis involves only the variance of measurement results

of the dealer and the access structure. In fact, the security of more general continuous-

variable CQ and QQ protocols can be analysed by using my techniques, i.e., transferring

the strong correlation to mode h and then compute the covariance matrix between mode

D and mode h, even if the resource state is not a continuous-variable cluster state.

To the best of my knowledge, the current work is the first one showing that quan-

tum secret sharing is feasible with finitely squeezed CV resources. A finitely squeezed

cluster state can be deterministically constructed by using only squeezed vacuum states

and linear optics, which are practically available resources in nowadays laboratory; our

work significantly lowers the required technological level for implementing quantum se-

cret sharing. However, there are more theoretical investigations have to be done before

our scheme is practically useful. An important remaining question is to determine if the

performance of quantum secret sharing is seriously worsened under the presence of envi-

ronmental noise and apparatus imperfections. Besides, the above calculations of secret

sharing rate are calculated at the asymptotic limit, which infinite rounds of state distri-

54


bution are assumed to have been conducted. In practice, the access structure and the

dealer only share finite number of cluster states, which the secret sharing rate would be

affected. Nevertheless, as I borrow the security analysis techniques from CV QKD, which

works well in noisy and finite-key circumstances, it is likely that the realistic performance

of quantum secret sharing can be analysed by using a similar formalism as presented in

this work.

55

Chapter 4

Motional States of Trapped Ions

Since it was realized that quantum algorithms can speed up complicated computational

tasks, such as factorizing a large integer, which cannot be efficiently performed by known

classical algorithms, building a quantum computer (QC) has become one of the ambitious

goals in modern physics [143, 5]. Among current proposals for physical implementations

of a QC, in many ways the ion trap proposal of Cirac and Zoller [50] seems the most

auspicious at the moment. By exploiting well-developed techniques from quantum optics

and atomic physics, entanglement of up to 14 ions [77, 136], high fidelity gates and

readout [139, 111], and long coherence time quantum memory for more than 10 s [103]

have been recently demonstrated. Simple quantum algorithms including Deutsch-Jozsa

algorithm [75] and quantum teleportation [151, 21], as well as the verification of the Bell

inequality [156], have been successfully realized in ion-trap QC.

In the conventional approach of ion trap QIP, two metastable electronic states are

employed as a two-level system (qubit) to encode quantum information, while the mo-

tional degree of freedom is acted only as a bus for transmitting quantum information.

Recently, directly implementing QIP on the motional states has received much atten-

tion. There are two reasons for this trend. Firstly, the advanced trapping technology

has reduced the heating rate on an trapped ion to lower than 0.1 quanta per millesec-

ond, which implies the motional coherent time is longer than 1 millisecond [36]. When

comparing with the characteristic time scale of ion motion, which is at the microsecond

range in state-of-the-art MHz trap, thousands of coherent operation can be conducted to

implement meaningful quantum information process. Secondly, each motional degree of

freedom of an ion is quantised as a harmonic oscillator that exhibits bosonic behaviour.

In contrast to the electronic states that behave like Fermions, the motional states could

function differently in quantum information processing and thus increase the capability of

ion trap systems. Additionally, more quantum information can be encoded in a harmonic

56

Chapter 4. Motional States of Trapped Ions

oscillator than in a qubit.

In my work, I have studied several aspects related to the quantised ion motion in

quantum information processing. In Ch. 5, I study the relation between ion transporta-

tion speed and dc Stark effect. From the relation, I deduce the threshold speed above

which quantum information will be affected by dc Stark effect during ion transport. In

Ch. 6 and Ch. 7, I introduce two architectures of universal bosonic simulator, which can

simulate the evolution of any optical system by manipulating the quantised ion motion.

In Ch. 8, I invent a new method to remove the unwanted thermal motion of a qubit by

collision. Such a method is one order of magnitude faster than the conventional laser

cooling methods.

Before discussing my works in detail, in this chapter I describe the basics of the ion

trap system that will be considered. In Sec. 4.1, I review the configuration of ion traps

and the mathematical tools for describing the quantised motion of trapped ions. In Sec.

4.2, I discuss the internal states and their interaction with laser fields. In Sec. 4.3,

I introduce a theory that describes the exact evolution of trapped ion motional states

under a general harmonic potential. This theory is the key component in my works,

because it allows us to develop processes to manipulate ion motion much faster than

conventional approaches.

4.1 Trap and Ion Motion

In spite of having similar internal structure to atoms, ions are more controllable due to

their net electric charge. Positively charged ions are usually employed in QIP due to

its stability against lost of charge. As the ion’s wave function is much shorter than the

typical length of trapping apparatus, an ion is usually treated as a point charge. The

necessary condition to stably trap a point charge is to construct an electric potential

V (~r), so that at the vicinity of the ion, the second derivative of the potential is positive

for all the three dimension, i.e.,

∂2

∂x2V > 0 ;

∂2

∂y2V > 0 ;

∂2

∂z2V > 0 . (4.1)

However, the potential created by any electrostatic distribution cannot simultaneously

satisfy all of the three conditions. This is because ∇2V = 0 due to the Maxwell’s equation

in vacuum.

The key to bypass this limitation is to use an oscillating, instead of a static, electric

field. Fig. 4.1 shows the design of the Paul trap, which its inventor Wolfgang Paul

57


rf

dc

++

+

Figure 4.1: Layout of Paul trap. Radio frequency potential is applied on the darkelectrodes, in order to trap the ions along the x axis. Positive electrostatic (dc) potentialis applied on the grey electrode segments, in order to trap the ions (little circles withpositive sign) in the the middle of the trap. More details can be referred to Fig. 3 inRef. [110].

was awarded the Nobel Prize due to this work [145]. The trap consists of four parallel

segmented electrodes. In the direction parallel to the electrodes (x direction in Fig. 4.1,

which is usually referred as “axial”), electrostatic potential is applied to stably confine

the charges. While in the directions perpendicular to the electrodes (y and z directions in

Fig. 4.1, which are usually referred as “radial”), a sinusoidal potential oscillating at radio

frequency (rf), ωrf, is applied to the electrodes of one of the diagonals (dark electrodes in

Fig. 4.1). The overall potential experienced by a point charge can be approximated as

V (~r) = Udcx2 + Uac cos(ωrft)(y

2 − z2) . (4.2)

For an ion with mass m experiencing such potential, the y-direction motion follows

the equation of motion:

my(t) = −Uac cos(ωrft)y(t) . (4.3)

The motion in the z-direction follows the same equation except the a π/2 phase shift

in the radio frequency potential. After scaling, this equation can be transformed to the

Mathieu equation [110]. For some range of Uac and ωrf, the solution of Mathieu equation

is bounded. Thus ions are stably trapped in the radial as well as the axial direction.

Before quantum information processing, ions are cooled to near the ground state

by Doppler and resolved sideband cooling [181]. For a singly trapped ion, around its

58


classical position the static axial trap potential and the effective radial trap potential can

be approximated by harmonic wells. The Hamiltonian of the ion motion is thus

Hm =p2x2m

+p2y2m

+p2z2m

+1

2mν2xx

2 +1

2mν2y y

2 +1

2mν2z z

2 , (4.4)

where νx depends on the axial static potential; νy and νz depend on Uac and ωrf [110].

By this Hamiltonian and canonical quantisation, the ion motion is quantised as three

independent harmonic oscillators. Each quanta of motional excitation in one direction

is regarded as a phonon. Because the phonons share the same form of Hamiltonian as

photons, which is also a harmonic oscillator, phonons are expected to exhibit bosonic

behaviours, and each motional degree of freedom is analogously referred as a mode.

Here I have introduced the quantisation of single ion motion. In future chapters, I will

discuss that if a trap contains more than one ions and they are close enough that Coulomb

interaction become significant, then the phonon mode has to be redefined as the collective

motion of multiple ions.

In experiments, the effective radial trapping frequency is about νy,z ≈ 2π × 10 MHz,

which is roughly one order of magnitude higher than the axial trapping frequency νx ≈2π × 1 MHz. In my work, I focus on the motional state on the axial direction, while

the radial motion is assumed to be tightly confined and would not be excited during

operation on the axial modes.

4.2 Internal Structure and Laser Operation

The type of ions involved in quantum information processing should have simple atomic

structure in order to be manipulable by laser operation. The singly charged alkaline

earth ions are particularly suitable because they have only one outer most electron, so

their electronic structure resembles that of a hydrogen atom.

In order to provide a long storage time of quantum information, encoding states should

be forbidden from dipole transition. Examples of suitable electronic states include the

4S and 3D states in 40Ca+ ion, which the decay time is about 1 second [90], and the

hyperfine states of 9Be+ ion, which the decay time is longer than centuries [66].

4.2.1 Transition

The transition between the encoding states requires non-dipole transition, such as quadrupole

transition or Raman transition [110]. In experiments, Raman transition is often employed

59


Figure 4.2: Energy levels of a Λ-type system (left side). The quantum informationencoding states |g〉 and |e〉 are dipole transition forbidden. The auxiliary state |d〉 isdipole transition allowed between both |g〉 and |e〉, with transition frequency ω1 and ω2

respectively. Under Raman transition caused by two detuned laser field, the electrontransits between |g〉 and |e〉 as in a two level system (right side), while the auxiliary stateis barely populated.

due to the strong coupling strength and the high stability of lasers. The implementation

of a Raman transition requires an auxiliary state that is dipole-transition allowed with

both encoding states; the energy level of an example of such system is shown in Fig. 4.2.

Two laser fields are applied that are both ∆ detuned from each of the dipole transition

frequency. If ∆ is much larger than the Rabi frequency of each laser field, the auxiliary

state is barely populated, while the electron transits between the encoding states just as

in a two-level system.

Unless specified, throughout this thesis I will consider the internal states of a trapped

ion as a two-level system, and the transition between the encoding states is effectively

driven by a single light field. For a field with frequency ω, wave vector ~k, and phase φ

with respect to some phase reference, the Hamiltonian is given by

H = ~Ω|e〉〈g| exp(i~k.r − ωt+ φ) + h.c. , (4.5)

where Ω is the Rabi frequency 1.

The spatial extent of the light field will couple with the ion motional states via the

term ~k.r, where r is the position vector operator of the ion position. The coupling

strength of a mode depends on (i) the intensity of light field, which determines the Rabi

frequency; (ii) the direction of the light field, contributes to the intersection angle in ~k.r;

and (iii) the frequency of the light field, ω, which would induce strong coupling when it

is in resonant with the transition and the mode frequency. In future discussions, I will

1For a Raman transition driven by two field both with Rabi frequency Ω0, the effective Rabi frequencyis Ω = Ω2

0/∆. The effective transition frequency, wave vector, and phase are ω1−ω2, ~k1−~k2, and φ2−φ1

respectively [110].

60


|g

|d

|e

|g

|d

|e

ed

No clickclick

ed

Figure 4.3: Left: If the electron is in |e〉, photons will be scattered. Right: If the electronis in |g〉, no photon is scattered.

consider the light field is tuned to be off-resonant from the radial mode, so these modes

will not be affected by the drive and could be neglected. In other words, I consider only

the ion’s axial motion, while the radial modes will remain in the ground state.

4.2.2 Measurement

Apart from transition, the electronic states can also be measured by applying a laser

field. The frequency of the field is tuned to resonant to the transition between |e〉 and an

excited state |d〉, i.e., ωed. If the state is in |e〉, the laser field will interact with the dipole

moment between |e〉 and |d〉, so some photon of the field is scattered. On the other hand,

if the state is in |g〉, the laser field would not interact with the electronic state. The

idea is shown schematically in Fig. 4.3. Therefore the electronic state can be projectively

measured by detecting the scattered photons (fluorescence) by nearby photon-detectors.

4.3 Ion Motion in Trap Potential

As previously described, each ion can be treated as a two-level system 2 with three

harmonic oscillators attached. For a large scale quantum information process, multiple

trapped ions are required to operate collaboratively. This is usually achieved by confining

a chain of ions along the axis of a single linear trap [50]. However, this method cannot

be scaled to involve much more than ten ions, otherwise the unwanted interaction may

become serious [181].

A scalable architecture ion trap quantum information processor is proposed in Refs.

[181, 99] (This architecture is referred as the Kielpinski-Monroe-Wineland (KMW) archi-

tecture throughout the thesis). The layout of this architecture is shown in Fig. 4.4. This

2More internal levels could be involved to form a qudit system, but each additional internal levelrequires an extra laser of different frequency, which is resources consuming in practice.

61


+

+

+ +

++

Interaction region

Memory region

Electrodes

Figure 4.4: Layout of scalable ion trap quantum information processor. Ions are initiallystored in storage traps (memory region). To conduct quantum logic operations, ions fromdifferent traps are transported to and combined into a single trap (interaction region).Both ion transportation and combination are implemented by varying the electrostaticpotential of the segmented electrodes. More details can be referred to Fig. 1 in Ref. [99].

architecture consists of interconnected traps, each of which is assigned for a specific func-

tion. In each trap serving as a memory, a chain of a small number of ions, say less than

ten, is stored. During quantum information processing, ions are transported to different

traps through linear traps and junctions. The ion transportation has been demonstrated

experimentally [155, 33, 175, 81, 30, 159]. When two chains of ions have to interact, they

are combined into a single interaction trap. After transportation or combination, the

ions can be cooled by sympathetic cooling to the ground motional state, so subsequent

quantum logic operations can be conducted precisely [22, 115].

The KMW architecture is regarded as a promising approach due to various advan-

tages. For example, each trap contains only a small number of ions, therefore the quality

of quantum logic operation is independent of the number of ions in the processor. Be-

sides, the operation in one trap is barely affecting the ions in other traps because they are

spatially well separated. Therefore logic operations can be conducted in parallel, hence

the number of operational cycles of a computational task can be reduced.

The key feature that allows these advantages is ion transportation. It can be imple-

mented by tuning the trapping potential through varying the voltage of the segmented

electrodes around the ions. The utility of a KMW quantum information processor could

be facilitated by rapid and precise manipulation of trapping potential. Such a degree of

control has been realized in recent experiments [33, 175]. Although the trapping potential

can be very complicated, it can be approximated by harmonic wells around the vicinity

62


of each ion. The quantised ion motion can then be well described by the evolution of

harmonic oscillators. In the following, I present the exact solution of a wave function

under a general harmonic potential. As we will see in future chapters, the exact solution

allows us to design quantum logic and operational processes that are fast and do not

induce additional heating.

4.3.1 Generalized Harmonic Oscillator

The most general harmonic oscillator consists of a harmonic well that both the potential

strength and the position of trap centre are time dependent. The wave function |ψ(t)〉obeys the Schrodinger equation

i~∂t|ψ(t)〉 = H|ψ(t)〉 ≡(

p2

2m+

1

2mν2(t)(x− s(t))2

)

|ψ(t)〉 . (4.6)

This generalized harmonic oscillator has been investigated in the context of driven Fock

states [118] and the variation of quadrature operators in a driven oscillator [100]. Here I

formulate it in the context of a trapped ion to give a clear physical understanding about

the quantum and classical evolution of the system.

Firstly, I decouple the classical motion and quantum fluctuation from the total mo-

tional state. Let us define the state of the quantum fluctuation, |χ(t)〉, as

|χ(t)〉 ≡ D†(xc, pc)|ψ(t)〉 , (4.7)

where xc(t) and pc(t) are real functions of time; the displacement operator is defined as

D(xc, pc) ≡ exp (i(xcp− pcx)/~) . (4.8)

The quantum fluctuation obeys the equation

i~∂t|χ(t)〉 =(

D†HD − iD†(∂tD))

|χ(t)〉 ≡ (H1 + H2)|χ(t)〉 . (4.9)

H1 is the collection of all terms containing the first order of position and momentum

operators, i.e., H1 = V p + F q, where

V =pcm

− xc ; F = pc +mν2(t)(

xc − s(t))

. (4.10)

If xc and pc are chosen respectively as the ion’s classical position and momentum, i.e.,

63


they obey the classical equation of motion

xc =pcm

; pc(t) = −mν2(t)(xc − s(t)) , (4.11)

then V and F , and thus H1 vanish. The dynamics of the quantum fluctuation is solely

determined by H2 that involves only the second order terms of position and momentum

operators, viz.

i∂t|χ(t)〉 = H2|χ(t)〉 ≡(

p2

2m+

1

2mν2(t)x2

)

|χ(t)〉 . (4.12)

H2 is the general Hamiltonian of a harmonic oscillator with time dependent well strength,

ν2(t), but a fixed well centre at x = 0. Now the action of the above operations becomes

clear: separating the classical and quantum attributes of motion. The classical motion

is completely described by the classical equation of motion, and it is controllable by

adjusting the well centre s(t). On the other hand, the quantum fluctuation of motion

evolves as a time dependent quantum harmonic oscillator that is independent of s(t). I

note that the above procedure of classical-quantum motion separation is also applicable

for other (non-harmonic) potentials.

The exact solution of a general time dependent harmonic oscillator was proposed by

Lewis and Riesenfeld. They deduce the solution by considering the dynamic invariant

operator [113, 45] :

I(t) = (bp−mbx)2

2m+

1

2mν20 x

2 = ~ν0

(

A†(t)A(t) +

1

2

)

. (4.13)

where b(t) is a dimensionless real auxiliary function that satisfies the equation

b+ ν2(t)b− ν20b3

= 0 . (4.14)

ν0 is a characteristic frequency of the problem that could be taken as the static harmonic

frequency before or after the potential variation. The operators A(t) and A†(t) are the

raising and lowering operators of the eigenstates, |λn, t〉, of I(t), i.e.

I(t)|λn, t〉 = λn|λn, t〉 , (4.15)

A(t)|λn, t〉 =√n|λn−1, t〉 , (4.16)

A†(t)|λn−1, t〉 =

√n|λn, t〉 , (4.17)

64


with λn being the corresponding eigenvalues.

The dynamic invariant is defined in such a way that its total time derivative vanishes,

i.e., its Heisenberg equation of motion becomes

d

dtI(t) ≡ i~∂tI(t) + [I(t), H(t)] = 0 . (4.18)

As the system evolves, the values of λn remain unchanged, and the eigenstates |λn, t〉 arealways orthogonal during the evolution, i.e.,

〈λm, t|(

i~∂t − H(t))

|λn, t〉 = 0 , if n 6= m . (4.19)

Then the evolution operator of quantum fluctuation from time t to t′ can be deduced as3

Uχ(t, t′) =

∞∑

n=0

e−i(n+ 12)(Θ(t)−Θ(t′))|λn, t〉〈λn, t′| . (4.20)

The phase Θ is chosen as

Θ(t) =

∫ t

0

ν0b2(t′′)

dt′′ , (4.21)

such that the states ei(n+12)Θ(t)|λn, t〉 are solutions of Eq. (4.12).

When the harmonic well is static, i.e. ν is a constant, the general real solution of

Eq. (4.14) is [113]

bstatic(t) =

√

ν0ν

√

cosh δ + sinh δ sin(2νt + ϕ) , (4.22)

where δ and ϕ are constant parameters. In my work, I am mainly interested in the

operations that the trapping potential are steady at the beginning and the end, i.e.,

ν(t < ti) = ν0 ; ν(t > tf) = νf (4.23)

where ti and tf are the starting and ending time of the operation. In general, the values

of δ and ϕ have to be determined by integrating Eq. (4.14), but for simplicity I can set

both δ = 0 and ϕ = 0 at the beginning, so that b(t < ti) = 1.

I pick the initial situation as a reference. The annihilation operator of the initial

oscillator is defined as

a =

√

mν02~

x+i√

2m~ν0p . (4.24)

3Throughout this thesis, an evolution operator is considered in the Schrodinger picture, while the“evolution operator” in the interaction picture is referred as the S-matrix that will be defined later.

65


Since the invariant operator I(t) is identical to H(t) at t = ti, we have

A(ti) = a ; (4.25)

and thus a|λn, ti〉 =√n|λn−1, ti〉 . (4.26)

After the operations, the lowering operator becomes [113]

A(tf) = η(tf)a+ ζ(tf)a† , (4.27)

where

η(t) =1

2

(

1

b+ b− i

b

ν0

)

, ζ(t) =1

2

(

1

b− b− i

b

ν0

)

. (4.28)

The absolute magnitudes of η(t) and ζ(t) satisfy the normalisation condition |η|2−|ζ |2 =1.

The action of the harmonic potential variation can be represented by the evolution

of the annihilation operator, i.e., U †χaUχ, which can be obtained from the relationship

between the raising operator and the annihilation operator. According to Eqs. (4.16)

and (4.26), we have

A(tf) = e−i(Θ(tf )−Θ(ti))Uχ(tf , ti)aU†χ(tf , ti) . (4.29)

By linearly combining the above equation and its complex conjugate, the evolution of

the annihilation operator in the Heisenberg picture is given by

U †χ(tf , ti)aUχ(tf , ti) = η∗(tf )e

−i(Θ(tf )−Θ(ti))a− ζ(tf)ei(Θ(tf )−Θ(ti))a† . (4.30)

Now I can combine the evolution of both the classical motion and quantum fluctuation

into a complete evolution of the wave function. Let us assume that both the harmonic

well and the ion are initially resting at the origin, i.e., s(ti) = 0 , xc(ti) = 0, and pc(ti) = 0.

By substituting the definition of Uχ into Eq. (4.7), the evolution operator of the total

wave function, i.e., U(t, ti)|ψ(ti)〉 = |ψ(t)〉, is given by

U(t, ti) = D(xc(t), pc(t))Uχ(t, ti) . (4.31)

Let us define a complex displacement β as

β(t) =

√

mν02~

xc(t) + i

√

1

2~mν0pc(t). (4.32)

66


Then the displacement operator in Eq. (4.8) can be rewritten as

D(xc, pc) ≡ D(β) = exp(βa† − β∗a†) . (4.33)

I note that now the annihilation operator corresponds to the total wave function instead

of the quantum fluctuation. The transformation of the annihilation operator is then given

by

U †(tf , ti)aU(tf , ti) = η∗(tf )e−i(Θ(tf )−Θ(ti))a− ζ(tf)e

i(Θ(tf )−Θ(ti))a† + β(tf) . (4.34)

If b(t) is known, a closed form of the classical displacement can be obtained as [100]

β(tf) = i

√

m

2~ν0

(

η∗(T )e−i(Θ(tf )−Θ(ti)) + ζ(T )ei(Θ(tf )−Θ(ti)))

∫ tf

ti

b(t)ν2(t)s(t)eiΘ(t)dt .

(4.35)

Eq. (4.34) shows that the most general harmonic potential causes only two effects on

the state: a squeezing operation, as indicated by the first two terms in the right hand

side; and a displacement operation, as represented by the last term in the right hand

side.

4.3.2 Interaction Picture

I have discussed the evolution of a general harmonic oscillator in the Heisenberg picture, it

may be useful to consider the evolution also in the interaction picture, which is commonly

employed in quantum optics. The interaction picture wave function is defined as

exp(

−iH0(t− ti)/~)

|ψI(t)〉 = |ψ(t)〉 , (4.36)

where H0 is an arbitrarily chosen reference harmonic oscillator

H0 ≡p2

2m+

1

2mν20 x

2 . (4.37)

The “evolution operator” of an interaction picture wave function is the S-matrix, i.e.,

|ψI(tf )〉 = S(tf , ti)|ψI(ti)〉 , (4.38)

where the S-matrix is defined as S(tf , ti) ≡ exp(iH0(tf − ti)/~)U(tf , ti). Therefore the

67


annihilation operator is transformed in the interaction picture as

a→ S†(tf , ti)aS(tf , ti) = U †(tf , ti)aU(tf , ti)eiν0(tf−ti) . (4.39)

From now on I denote the notation “→” as the above transformation.

By combining Eqs. (4.39) and (4.34), the interaction picture evolution of the annihi-

lation operator is given by

a→ η∗(tf )e−i(Θ(tf )−Θ(ti)−ν0(tf−ti))a− ζ(tf)e

i(Θ(tf )−Θ(ti)+ν0(tf−ti))a† + βeiν0(tf−ti) . (4.40)

68

Chapter 5

Decoherence Induced by dc Electric

field During Ion Transport

5.1 Speed of ion trap quantum computer

In spite of the rapid advancement of the ion trap system, we are still far from having

a quantum computer (QC) with the computing power higher than (or even comparable

to) its classical counterparts. Apart from the problem that only a small number of

entangled qubits have been realised, the speed of quantum operations is another issue

that limits the clock rate of an ion trap QC. For example, let us consider a QC is built

to run the Shor algorithm. It would be of great practical interest only if it could break a

RSA classical cryptography code in a short time, say a few hours. Then each quantum

logic gate has to be performed at the time scale of µs 1 [65]. When employing fault-

tolerant techniques, each quantum logic gate consists of several concatenated rounds of

physical quantum operations on ion qubits, including transportation, cooling, and laser

interaction [181, 99]. Assessing the true time specifications for a QC in a complicated

problem depends on paradigm choices, such as circuit-based versus measurement-based

QC, the error correction code used, and the implementation of the algorithm to be

performed. But we can assert quantum computation is useful and promising if the speed

of each physical quantum operation is of nanosecond scale [164].

In this chapter, I present my work with Daniel James in investigating the effect of

direct current (dc) Stark shifts during ion trap QC operation according to the KMW

architecture [181, 99] discussed in Sec. 4.3 . During quantum computing, ions are moved

from the storage region to the interaction region, and transported back to storage region

1As shown on Table 1 of [65], the most efficient quantum algorithm for factorizing a 640 bit numberrequires 3000× (640)2 = 1.2× 109 quantum operations.

69

Chapter 5. Decoherence Induced by dc Electric field During Ion Transport

after operations. The ions can be transported by changing the electric potential of each

trap that induces an effective non-equilibrium electric field.

Fast transportation of an ion requires a large electric field, which will result in a va-

riety of potentially detrimental effects. One problem is that a large electric field is less

stable; the field fluctuations will heat up the ion to motional excited states [155]. Cycles

of sympathetic cooling are required to bring the ion back to its motional ground state

for precise logic operations, but the operation is time-consuming and thus not preferable

in high-speed quantum computation. The motional heating effect is anticipated to be

reduced by improving experimental techniques, such as using surface traps and coating

the electrodes of the trap [155], or transporting the ions under trajectories with mini-

mal vibrational quanta excitation [168]. Besides, some proposals for entanglement gates

remain effective even though the ion has a small motional excitation [163, 91].

More seriously, a large electric field will induce a dc Stark effect onto the internal

electronic states of ions. Due to the detuning of energy levels and mixing of eigenstates,

information-encoded quantum state will be altered by being (i) phase-shifted, and (ii)

excited out of the computational basis. Although this issue was first discussed more than

10 years ago by Wineland et al. (see Ref. [181] p.310), its significance was neglected

at that time due to the low operational speed they envisioned. This effect will become

important as the trapped ion shuttling speed is becoming faster; ignoring this effect may

cause decoherence on the quantum information. The aim of my work is to find out the

relation between the total effects by dc Stark shift, the size of trap, trajectory and total

time of the flight of ion qubits. From the relation, I define the ‘threshold speed’ of ion

transportation, above which the influence of dc Stark effect becomes significant.

This chapter is organized as follows. In Sec. 5.2, I use the tools presented in Sec. 4.3

to calculate the classical trajectory of an ion being transported by a displacing harmonic

well. Because the ion’s classical motion is determined only by the dc electric field, I could

obtain from the acceleration the net electric field experienced by the ion. In Sec. 5.3,

the total phase-shift induced by dc Stark effect is calculated. In Sec. 5.4, I minimize the

phase-shift with respect to the ion trajectory. A threshold speed is deduced from the

relation between the minimum phase shift and the optimized time of flight. In Sec. 5.5,

I calculate the threshold speed for various choices of ion qubits. In Sec. 5.6, I discuss

the significance of state excitation caused by dc Stark effect. I summarize the results in

Sec. 5.7 with some discussion. I note that most of the material in this chapter has been

published as Ref [106]. For simplicity, I do not consider the junctions where ions are

transferred between different traps; the ion is assumed to move along a linear trap only.

70


5.2 Motion of ion

In the axial direction, the ion is weakly trapped by an electrostatic field. Close to a

minimum of the axial electrostatic potential, the Hamiltonian of the system can be ap-

proximated by a harmonic oscillator 2. During shuttling, the strength of the electrostatic

field is changed so that the potential well is displaced. In order to avoid parametric ex-

citation that complicates the problem, let us suppose the the strength of harmonic well

is tuned to be constant throughout the process, i.e., νx(t) = ν0. The time-dependent

Hamiltonian is given by

HM(t) ≡ p2

2m+

1

2mν20 [x− s(t)]2 . (5.1)

Let the transportation lasts from t = 0 to t = T . The general evolution operator of

the Hamiltonian in Eq. (5.1) is given by Eq. (4.31). The constant trap frequency gives a

constant auxiliary function, i.e., b(t) = 1, and as a result Uχ is reduced to a phase-shift

operator, i.e.,

U †χ(t)aU

†χ(t) = ae−iν0t . (5.2)

If the harmonic potential is centered initially at x(0) = 0 and the ion is prepared in

the ground motional state, i.e.|Ψ(0)〉 = |0〉, Eqs. (4.31) and (5.2) implies that the ion

remains in a coherent state throughout the whole process, i.e. |Ψ(t)〉 = |α(t)〉. The

displacement α(t) can be obtained from Eq.(4.35) with b(t) = 1 and ν(t) = ν0. After an

integration-by-parts, the displacement can be found as

α(t) ≡√

mν02~

(

s(t)− e−iν0t

∫ t

0

s(t1)eiν0t1dt1

)

. (5.3)

The first term on the right hand side represents the position of the potential well centre,

while the second term, which represents the displacement of the ion with respect to the

well, contributes to the motional heating when ignored. I note that a s(t) can always be

found to produce any final displacement (classical position and momentum) [168, 46].

Time variation of the expectation value of the ion position (classical trajectory of the

ion), q(t), can be calculated as

q(t) = s(t)−∫ t

0

s(t1) cos [ν0(t− t1)] dt1 . (5.4)

2Here I only consider harmonic potentials, more general trapping potential and detailed dynamics oftrapped ion are discussed in [168].

71


The net dc electric field experienced by the ion, ~ξ = ξ~ex where ~ex is the unit vector along

x direction, is directly proportional to the ion’s classical acceleration (second derivative

of the position expectation value) by the Newton’s third law, viz.,

ξ(t) =m

eq(t) =

mν20e

∫ t

0

s(t1) cos[ν0(t1 − t)]dt1 . (5.5)

5.3 Phase shift due to dc Stark effect

In this section, I study the total phase shift during the transportation process. The effect

of the applied electric field on the ion’s internal structure is described by an additional

Hamiltonian HStark = −~d · ~ξ, where ~d is the dipole operator. Since the electric field varies

much slower than the electronic states evolution, which the time scale is characterised

by 1/ωnm, the dc Stark shift energy at time t can be obtained by the time independent

perturbation theory [6], viz.,

En(t) ≈ E(0)n + E(2)

n (t); E(2)n (t) =

∑

m6=n

|eξ(t)〈m|x|n〉|2~ωnm

(5.6)

where electric field points to x direction only; ~ωnm ≡ ~ωn−~ωm is the energy difference

between internal states |n〉 and |m〉.Suppose |f〉 and |i〉 are the computational basis states 3. The ion is initially encoded

with some quantum information as the state α|i〉 + β|f〉. Because the dc Stark energy

of each internal state is different, a relative phase between the computational states will

be induced after transportation, i.e.,

α|i〉+ β|f〉 → α|i〉+ βeiφ|f〉 ,

up to an unimportant global phase. According to Eqs. (5.5) and (5.6), the extra phase

factor φ is given by

φ ≡∫ T

0

E(2)i (t)−E

(2)f (t)

~dt =

m2

~

(

∑

m6=i

|〈m|x|i〉|2~ωim

−∑

m6=f

|〈m|x|f〉|2~ωfm

)

ζ [q(t)] , (5.7)

where

ζ [q(t)] =

∫ T

0

q(t)2dt . (5.8)

All terms except ζ depend on only the atomic structure of the ion, so a good choice of ion

3The relative phase due to the energy difference of the states is eliminated in the interaction picture.

72


qubit would give a small magnitude of these terms. ζ is a functional of the ion trajectory,

q(t), which is independent of the choice of ion species and the computational states.

5.4 Minimum possible phase shift

As shown in Eqs. (5.7) and (5.8), the phase shift is linearly proportional to the trajectory

functional ζ . Among all the possible trajectories, there is an optimal one, q0, which

produces the minimum ζ , and hence the minimum phase shift, φmin. I note that ζ is

independent of the ion species, so its minimum value, ζ [q0], is only a function of the

transportation length, L, and the time of the flight, T .

The minimum ζ [q0] can by found by using calculus of variation (see, e.g. [7]). I first

set the speed of the ion, γ(t), as an independent parameter function. Its relationship

with the ion position, i.e.,

γ(t) = q(t), (5.9)

will be treated as a constraint of the minimisation. Then ζ can be rewritten as

ζ[q, q, γ, γ, t] =

∫ T

0

γ2(t) + µ(t)[q(t)− γ(t)]

dt , (5.10)

where µ(t) is the Lagrange multiplier for the constraint in Eq. (5.9). Although the

minimisation of ζ and ζ are equivalent, it is beneficial to conduct calculus of variation

on ζ because it is a functional of q, γ, and their first order time derivative only (i.e. no

high order derivatives).

The Euler equations with respect to q and γ are given by

µ(t)− 2γ(t) = 0 (5.11)

µ(t) = 0 . (5.12)

Incorporating these equations and the constraint Eq. (5.9), we find q0 obeys

....q0 (t) = 0 . (5.13)

I assume the ion is displaced by distance L after the process, the initial and final

position can be set as q(0) = 0 and q(T ) = L. Additionally, I require the ion remains in

the motional ground state before and after the shuttling, so the initial and final velocity

both vanish, i.e. q(0) = q(T ) = 0. Putting these conditions into Eq. (5.13), the optimal

73


trajectory can be found as

q0(t) = L

(

−2t3

T 3+ 3

t2

T 2

)

, (5.14)

which gives ζ(q0) = 12L2/T 3. Hence the minimum phase shift is

φmin =12m2L2

~T 3

(

∑

m6=i

|〈m|x|i〉|2~ωim

−∑

m6=f

|〈m|x|f〉|2~ωfm

)

. (5.15)

Contrary to the adiabatic transportation [156], I note that the ion does not stay the

equilibrium position of the potential well s(t) throughout the transportation. As expected

by the classical intuition, the ion ‘sloshes’ when the well is displaced. Transporting the

ion in the desired trajectory has to be achieved by tuning the trapping electric field

carefully to balance this sloshing. The position of the well can be obtained by combining

Eq. (5.4) with its second derivative. The displacement of the well s0(t) which produces

the optimal trajectory of ion is given by:

s0(t) = q0(t) +qo(t)

ν20= L

(

−2t3

T 3+ 3

t2

T 2− 12

t

ν20T3+

6

(ν0T )2

)

. (5.16)

Both q0(t) and s0(t) are shown schematically in Fig. 5.1.

The position of the potential well jumps sharply at t = 0 and t = T . This means that

the trapping electric fields are sharply changed at the beginning and at the end. However,

the sudden variation of electric field does not cause a significant dc Stark effect. This is

because the energy perturbation, within the dipole approximation, only depends on the

strength of electric field but not its time derivative.

Although the phase shift is minimum when the ion travels in the optimal trajectory,

but trajectories other than q0 may be employed due to experimental convenience. Here

I investigate the robustness of the optimality, by studying if the value of ζ dramatically

increases when the ion takes another trajectory. Firstly, I consider a more experimental

realizable trajectory that the electric field is tuned gradually, i.e. no sharp jumps of

electric potential. I construct a trajectory that is a fifth order of time, so that there

are six parameters to incorporate the constraints of the continuity of ion motion, i.e.,

q(0) = 0, q(T ) = L, q(0) = q(T ) = 0, as well as the continuity of harmonic well position,

i.e., s(0) = 0, s(T ) = L. Such an ion trajectory is given by

q(t) = L(6t5/T 5 − 15t4T 4 + 10t3/T 3) . (5.17)

74


Figure 5.1: Time variation of the optimal trajectory of ion, q0(t) (solid line), and theoptimal trajectory of potential well, s0(t), for trapping frequencies ν0 = 3/T (dashed),ν0 = 5/T (dotted), and ν0 = 10/T (dot-dashed).

This trajectory gives ζ = 17.1L2/T 3, which is larger than ζ [q0] but remains in the same

order of magnitude.

Secondly, I consider the trajectory of the experimental setting of Rowe et al. [156],

where the location of the trapping potential sR varies as

sR(t) = L sin2

(

πt

2T

)

, (5.18)

and the frequency of the potential well is νR = 2π × 2.9 MHz. By obtaining the ion

trajectory qR from Eq. (5.4), ζ is found to be ζ(qR) = 24.3L2/T 3, which is only about

a double of ζ [q0]. The ζ in these two examples are at the same order as ζ [q0], therefore

φmin is a useful reference for the phase shift induced during ion transportation.

I note that if the anharmonicity of the potential well is ignored, following the above

trajectories an ion can be transported in an arbitrary short time. The transportation

would induce minimal final motional excitation even though the speed is much higher

than the adiabatic limit. 4

4To compare the adiabatic and non-adiabatic transportation speed, let us consider the adiabatictrajectory s(t) = tL/T . Adiabaticity requires the motional excitation about the trap centre to be smallat any time during the transportation, i.e., |α(t) − s(t)|2 ≪ 1. According to Eq. (5.3), the adiabaticity

requirement can be written as 2m~ν0

L2

T 2 ≪ 1. As an example, for a trap with length L ≈ 100µm, a 9Be+ion in a MHz trap has to be transported at a time T ≫ 0.66 ms. In a recent experiment, a trajectory

75


5.5 Threshold speed of transporting ion qubits

In this section, I calculate the threshold speed of ion transportation, above which φmin

becomes significant, i.e., the magnitude is too large to be corrected by quantum error

correction. Since the threshold error rate of quantum error correction codes are about

1% (see e.g. [149]), I set the upper limit of φmin as π/100. I assume the length of the

trap being travelled is L = 100µm, which matches setting of current experimental setup

[156].

Two popular types of ion qubits are investigated. The first one is 40Ca+ ion [151],

where the computational basis states are electronic states |S1/2,MJ = −1/2〉 = |i〉 and

|D5/2MJ = −1/2〉 = |f〉. The second type is 9Be+ ion [103, 21, 156], where the compu-

tational states are hyperfine states |F = 2, m = 0〉 = |i〉 and |F = 1,MF = 1〉 = |f〉.Now I calculate the term inside the bracket in Eq. (5.15). Values of the matrix

elements can be calculated from the tabulated parameters in Ref. [8]. I demonstrate

here the calculation of the terms belonging to |i〉 while the procedure for |f〉 is similar.

Let the state |i〉 belongs to an energy level k. It can be expressed as a superposition of

states with definite magnetic quantum number |r〉, i.e.,

|i〉 =∑

r∈kAir|r〉 . (5.19)

I pick the intermediate states |m〉 as the energy eigenstates with a definite magnetic

quantum number, so the summation of |m〉 involves summing all the states in a particular

energy level l, and then summing over all energy levels. The transition energy ~ωim is

the energy difference between levels k and l, which the value can be found in Ref. [8].

When calculating the matrix elements of the dipole operator, I extract the reduced

matrix elements as

∑

m∈l|〈m|x|i〉|2 = |〈k||Q1||l〉|2

∑

r∈k

∑

m∈l|Air|2|Cmr|2 , (5.20)

where Q1 is the rank 1 irreducible tensor operator, |〈k||Q1||l〉|2 is the reduced matrix

element 5 between energy levels k and l, Cmr is the Clebsch-Gordan coefficients between

|m〉 and |r〉 6. The reduced matrix elements can be calculated from the line strength Slk

derived from Eq. (5.3) has been employed in transporting a 9Be+ ion across a 370 µm trap [33]. Whilethe final excitation is insignificant (about 0.1 quanta), the transportation is conducted in 8 µs, which istwo orders of magnitude faster than the adiabatic limit.

5Not to confuse with the reduced density matrix, which is the quantum state of a subsystem aftertracing out some degree of freedom of a quantum system.

6Expressions of the Clebsch-Gordan coefficients are given in, e.g. Appendix 4, p.1000 in Ref. [9].

76


provided in Ref. [8]. By definition,

Slk = gk∑

m∈l〈p|~d|m〉 · 〈m|~d|p〉 = e2gk|〈k||Q1||l〉|2 , (5.21)

where gk is the degeneracy of the level k. The last relation is derived from the orthonor-

mality condition of Clebsch-Gordan coefficients.

For a 40Ca+ qubit, only the states |m〉 = |P3/2,MJ = −1/2〉 and |P1/2,MJ = −1/2〉can yield non-zero dipole matrix elements. By using the formalism mentioned above, I

find the minimum phase is

φCamin = 9.86× 10−18 [L

2]

[T 3], (5.22)

where the square bracket denotes the value of quantities in S.I. unit. Both computational

states of a 9Be+ qubit are hyperfine states of the ground electronic state, so |m〉 are stateson P1/2 and P3/2 levels only. I find the minimum phase is

φBemin = 2.6× 10−25 [L

2]

[T 3]. (5.23)

For L = 100 µm and φmin . π/100, the threshold time of flight for ion qubits are

TCamin,100µm & 14.6 ns (5.24)

TBemin,100µm & 0.044 ns . (5.25)

TBemin,100µm is 3 orders of magnitude smaller than TCa

min,100µm because the energy between

two qubit states, EZ , is Zeeman energy that is small when compared with the energy

difference between atomic energy levels, EA. If no magnetic field is applied and the tiny

hyperfine splitting is neglected, the value of the bracket in Eq. (5.15) vanishes since both

qubit states are atomic ground states. The first non-vanishing term would be suppressed

by a factor of EZ/EA ≈ 10−6. Thus the phase shift of a 9Be+ qubit is much smaller than

that of a 40Ca+ qubit.

5.6 Non-encoding State Excitation

Apart from shifting the phase, the dc electric field also excites the electron into states

outside the computational basis. I start the analysis by writing the dc Stark effect

77


Hamiltonian in the interaction picture as

VI(t) =m

eVrnq(t)e

iωrnt|r〉〈n| , (5.26)

where

〈r|HStark|n〉 ≡m

eVrnq(t) = −m

e〈r|~d · x|n〉q(t) . (5.27)

By applying time dependent perturbation theory, the first three terms of the Dyson series

of the propagator UI(t) are given by

UI(t) ≈ I − i

~

m

e

∑

rn

Vrn

∫ t

0

q(t′)eiωrnt′dt′|r〉〈n|

− 1

~2

m2

e2

∑

rn

∑

r′

Vrr′Vr′n

∫ t

0

∫ t′

0

q(t′′)q(t′)ei(ωr′nt′′+ωrr′ t

′)dt′′dt′|r〉〈n| .(5.28)

The two most significant terms are: the first order term with ωr 6= ωn7; and the second

order term with ωr = ωn8.

Although the time integrals cannot be solved without the exact form of q(t), the

significance of each term can be compared by the estimated magnitude. After simple

integration by parts, the integral in the first order term in Eq. (5.28) will be

∫ T

0

q(t)eiωrntdt =−iωrn

(

q(T )eiωrnT − q(0))

+i

ωrn

∫ T

0

...q (t)eiωrntdt . (5.29)

Since the typical order of atomic transition frequency ωrn is 1015 s−1 and the timescale

of ion transport is about 10−9 s, the last term on the right hand side can be neglected as

it is much smaller than the term in the left hand side 9. As I am interested only in the

order of magnitude, it is appropriate to consider the bracket in the right hand side has

the same magnitude as L/T 2.

The magnitude of the second order term in Eq. (5.28) is estimated by a trick. It can

be recognised that the term for |r〉 = |n〉 is responsible for the phase-shift that is studiedin Sec. 5.5. Therefore we know the time integral can be approximated by iζ [q]/ωr′n,

which the value is at the order of 10L2/T 3. Here I consider the fact that the dominating

Vr′n are those of the low lying energy states, and the matrix elements of the states in the

same energy levels are in the same order of magnitude. The ratio of the first and second

7The states in the same energy level have the same parity and so Vnn = 08Because ωrr′ +ωr′n = 0 is required, otherwise there will be a rapidly oscillating term in the integral

and the contribution of the term is reduced by an order of perturbation.9assuming

...q ≈ q/T , the right hand side term is 1/ωrnT smaller, see Ref. [6]

78


order term can be estimated as

first order

second order≈ ~eT

10m|Vrn|L. (5.30)

For a qubit moving in a 100 µm trap within 10 ns, the above ratio is about 10−4. Therefore

the second order term in Eq. (5.28) dominates the state excitation.

Let us consider the dc Stark effect has excited the electron to a non-computational

state |bi〉, i.e.,UI(t)|Ψ〉 =

√

1−∑

i

|ǫi|2|Ψ〉+∑

i

ǫi|bi〉 . (5.31)

The probability of exciting a non-computational state is∑

i |ǫi|2. For a 40Ca+ qubit, the

amplitude of the second order term in Eq. (5.28) is the same order of magnitude as the

phase shift φ, so the probability of non-computational state excitation is about φ2, which

is much smaller than the phase error φ. Therefore the threshold speed of a calcium ion

QC is determined by the phase-shift effect, i.e. Eq. (5.22).

For a 9Be+ qubit, the second order term in Eq. (5.28) vanishes if hyperfine splitting

is neglected. This is because both computational states consist of only one orbital state,

|l = 0, ml = 0〉, i.e.

|2, 0〉F =

(

1√2|32,−1

2〉I |

1

2,1

2〉S +

1√2|32,1

2〉I |

1

2,−1

2〉S)

⊗ |l = 0, ml = 0〉 , (5.32)

|1, 1〉F =

(√3

2|32,3

2〉I |

1

2,−1

2〉S − 1

2|32,1

2〉I |

1

2,1

2〉S)

⊗ |l = 0, ml = 0〉 , (5.33)

where the first and second number of a state denotes respectively the angular momentum

and magnetic quantum number; the state with the subscript I (S) corresponds to the

nuclear (electronic) spin. Since the orbital state has no degeneracy, non-computational

states are off-resonant and barely excited by dc Stark effect.

If hyperfine splitting is included, the second order term in Eq. (5.28) is finite but its

magnitude is suppressed by a factor of EZ/EA ≈ 10−6. According to previous analysis,

the first order term may then become dominant. However, this term could be suppressed

by tuning the trajectory of the ion qubit. Let us recall that I have approximated the

bracket in Eq. (5.29) by the value L/T 2. However, if a trajectory is chosen so that the

initial and final acceleration of the ion qubit are both zero, then the bracket vanishes and

the first order will be reduced by a factor of (ωrnT )−1, which is about 10−6 for T = 1 ns.

Although the optimal trajectory q0(t) does not satisfy this criteria, a modified trajectory

can be constructed without significantly sacrificing the optimality. One possible modi-

79


fication is to include two buffer periods, each lasts for time T ′, at the beginning and at

the end of the transportation. During the beginning buffer period, the ion acceleration

is increased from zero to q0(0); similarly in the ending buffer period, the acceleration is

decreased from q0(T ) to 0 at the end of the flight, while the remainder of the trajectory

still follows q0(t). I find that if T ′ is much larger than 1/ωrn but much smaller than

T , then the first order term in Eq. (5.28) is greatly suppressed, while the phase shift is

scaled by only a factor of order unity.

Therefore the second order term in Eq. (5.28) of a 9Be+ qubit remains dominant.

Just as the 40Ca+ qubit, the amplitude of non-computational states is at the same order

as the phase shift of the computational states, so the probability of excitation is much

smaller than the phase-shift error. Therefore, the threshold speed of 9Be+ qubit is still

given by Eq. (5.25).

5.7 Summary and Discussion

In this chapter, I have studied the influence of dc Stark effects on the quantum information

encoded in an ion qubit. I find that phase-shift is the most significant error, while non-

computational state excitation is a less important effect. The magnitude of the phase-shift

scales as quadratically as size of traps and inversely cubic as the shuttling time. For an

ion qubit being transported across a 100 µm trap, the threshold time of flight of a 40Ca+

ion, Tmin,100µm, is about 10 ns unless the induced phase is larger than 1%. On the other

hand, a 9Be+ qubit is more resistant to the dc Stark effect. I find that the induced phase

is 6 order smaller than that of the 40Ca+ ion, so the threshold transportation time of a9Be+ qubit is as short as 50 ps.

In principle, the average Stark shift might be assessed by Ramsey interferometry, and

then corrected by unitary transformations. In practice, however, the ion is undergoing

a complex trajectory involving acceleration and deceleration, moving in straight lines,

turning around bends and through junctions, disengaging the individual ion from the

storage register and the logic trap; all of these effects will be too complicated to track

and calculate the Stark shift accurately. The uncertainties will turn the phase-shift

and excitation into errors. Quantum error correction will perforce be needed, and the

requirements of fault-tolerant quantum computing (in particular, ensuring the error be

below some threshold) will place a speed limit on the operation of the QC. Magnitudes

of the errors depend on the setup of the ion trap system. In any case, our result is a

useful reference to the speed limit. As an illustration, suppose the overall Stark shifts can

be evaluated, with accurately tracking the trajectory, well-controlled electric field, and

80


other very precise experimental techniques, up to 90% accuracy for a particular ion trap

QC. According to Eq. (5.22) and (5.23), this 10% of apparatus uncertainty will impose

a transportation time limit that is about half of the threshold time I have calculated 10.

A possible way to lower this threshold is to reduce the size of the trap. But this

method is inefficient because Tmin only scales at L2/3, and the reduction of trap size

may cause more serious heating on the qubits. Another method is to encode quantum

information into the decoherence free subspace of two ions [114, 99, 153], i.e. |0〉 → |i1f2〉,|1〉 → |f1i2〉. If the ions are transported in the same trajectory, the phase-shift on each

ion would be decoupled from the quantum information encoded in the logical qubit. In

addition, there are alternative scalable ion trap QC architectures that require much fewer

two qubit operations, hence fewer ion transportation (the measurement-based ion trap

QC, c.f. [164, 147]), or even no transportation of ions (the ion-photon network model,

c.f. [32, 133, 55]). The dc Stark effect is thus insignificant in these architectures.

10Due to the T 3 dependence of the induced phase, 3√0.1 ≈ 46%.

81

Chapter 6

Ion Trap Bosonic Simulator 1:

Multiple Ions in Single Trap

Apart from the dramatical speed-up of quantum algorithms running on a universal quan-

tum computer (UQC), we can also take advantage of the capability of quantum mechanics

by constructing a less versatile,i.e., without implementing the full set of universal logical

gates with accuracy above the fault tolerant threshold, quantum device to perform spe-

cific tasks. One such proposal is to use well-controlled quantum systems to simulate other

physical systems that are too complicated, e.g. due to the exponentially large Hilbert

space required, to be simulated on classical computers [61]. Because the precision re-

quirement of a quantum simulator is less stringent than a UQC [41], quantum simulation

is conceived to be realized by technology close to the state-of-the-art. Due to its techno-

logical maturity, the trapped ion system has been proposed and employed as the platform

for physical simulations of, for example, relativistic quantum effects [70], quantum phase

transitions [72], and the evolution of open systems [169] 1. In the following two chapters,

I will propose two ion trap architectures that simulate bosonic quantum systems.

6.1 Introduction

Coherently manipulated photons have been proposed to be a good candidate for test-

ing the foundation of quantum mechanics [178], performing quantum computations [101,

144], conducting high precision measurements [140], and many other applications. How-

ever, because of poor sources, detection inefficiencies, and weak photon-photon interac-

tions, implementing these proposals for large-scale devices is very difficult. It would be

1A good review of trapped-ion quantum simulator experiments is Ref. [92].

82

Chapter 6. Ion Trap Bosonic Simulator 1: Multiple Ions in Single Trap

fundamentally and practically interesting if we could build a quantum simulator of pho-

tons by using another well-controlled system with the same (bosonic) behaviour. More

specifically, a universal bosonic simulator (UBS) should be able to reproduce the evolu-

tion of a bosonic system under the most general form of Hamiltonian. This requirement

is not too stringent as the evolution can be approximated to arbitrary accuracy by a

sequence of basic operators that belong to a universal set [116]. Lloyd and Braunstein

[117] suggested that the simplest universal set of basic operators comprised: all single

mode linear operators; at least one multi-mode operator; and at least one nonlinear el-

ement. Efficiently performing only these basic operations is necessary and sufficient for

implementing a UBS.

Ion traps are a suitable candidate for implementing a UBS, in which a high degree

of controllability has been demonstrated [78]. The motion of laser-cooled ions is quan-

tum in nature, and the excitations of the motional states, i.e. phonons, exhibit bosonic

behaviour. The collective displacement and momentum of the ions are analogous to the

quadratures of light fields. Any arbitrary motional state can be created by combining

techniques such as sideband transition [181], parametric amplification [80], and adiabatic

passage [49]; in particular, the creation of Gaussian states [127] and non-classical states

[127, 134] from the ground state have been experimentally demonstrated. When applied

to non-ground states, some of these techniques can achieve single phonon linear or nonlin-

ear operations. Interaction between phonon modes at the few-quanta level has also been

observed. For example, nonlinear beam splitting on a single ion has been performed by

applying a Raman field [112]; coupling two phonon modes have also been demonstrated

through the Coulomb interaction between two separately trapped ions [36, 79], or two

ions in the same trap [154, 141].

In this chapter, I revisit the idea of trapped-ion bosonic simulation that was first

proposed by Wineland et. al. [180, 112]. Their proposal consists of only one trapped ion,

which could provide at most three motional modes for simulating three bosonic modes.

Aiming for a larger scale of simulation, I extend their idea by considering a trap with

multiple ions. The number of bosonic modes available in the simulator is proportional to

the number of ions. I will show that the basic operators of Hamiltonian can be realised

by applying finely-tuned laser field to couple the motional and the internal states of

ions. As an example, I outline the procedure for implementing Hong-Ou-Mandel effect

on phonons.

83


xi+1xi ν0

Figure 6.1: Layout of the bosonic simulator consisting of multiple ions in a single har-monic well, with trap frequency ν0. The ions are aligned along the axial direction (x inthe figure), while radial motion is strongly confined (not shown in the figure). Before thesimulation, the ions are cooled to the ground motional state, and the classical position ofeach ion is the equilibrium point between the harmonic trapping force coming from theelectrodes and the mutual Coulomb force between ions.

6.2 Layout of the system

I consider the UBS to be composed of N ions trapped in a linear trap. All the ions are

prepared in the ground electronic state. The ions are weakly trapped along the axial

direction, while the effective radial potential is strong enough that the ions are aligned

linearly but not in zigzag configuration. The layout of our system is shown schematically

in Fig. 6.1. The Hamiltonian that governs the ion motional states can be approximated

as [90]

H =

N∑

i=1

p2i2m

+ V ≡N∑

i=1

(

p2i2m

+1

2mν20 x

2i

)

+

N∑

i>j

e2

4πǫ0

1

xi − xj, (6.1)

where pi and xi are the momentum and position operator of the ith ion; ν0 is the harmonic

frequency of the trap; V is the total potential that includes the harmonic trap potential

and the Coulomb potential between ions.

The position operator xi can be expressed as the sum of the classical equilibrium

position, x(0)i , and the position operator of the quantum fluctuation, qi

2. The spread

of quantum fluctuation is much smaller than the classical separation between ions at

low temperature, i.e. 〈q2i 〉/|x(0)i − x(0)j 6=i| ≪ 1. Let us Taylor-expand the Hamiltonian in

Eq. (6.1) and apply the quadratic approximation, i.e., collecting terms up to the second

order of q’s, the Hamiltonian will become a coupled quantum harmonic oscillator, viz.

H ≈N∑

i=1

p2i2m

+ V2 ≡N∑

i=1

p2i2m

+

N∑

i,j

1

2qiqj

∂2V

∂x(0)i ∂x

(0)j

∣

∣

∣

qi=qj=0, (6.2)

where V2 is the quadratic approximated Hamiltonian 3. After diagonalizing V2, the Hamil-

2For simplicity, I have neglected the identity operator associated to the classical position. Formaltreatment of the separation of classical and quantum contribution can be given in Ref. [90] or Sec. 4.3

3the partial derivative of V is constructed by treating V and q in Eq. (6.2) as scalar function and

84


tonian in Eq. (6.2) becomes a sum of independent harmonic oscillators that represent the

collective motion of the ions, i.e., phonon modes, viz.

H =N∑

k=1

P 2k

2m+

N∑

k=1

1

2mν2kQ

2k =

N∑

k=1

~νk(a†kak +

1

2) . (6.3)

The position operator of the kth phonon mode is given by

Qk =

N∑

i=1

αkiqi , (6.4)

where (αk1, αk2, . . . , αkn)T is the kth eigenvector of the matrix ∂2V

∂x(0)i ∂x

(0)j

∣

∣

∣

qi=qj=0; Pk is the

conjugate momentum of Qk; mν2k are the eigenvalues of ∂2V

∂x(0)i ∂x

(0)j

∣

∣

∣

qi=qj=0, where νk is the

frequency of the kth mode. If there are N phonon modes in this system, N bosonic

modes can be simulated. The annihilation and creation operators of the kth phonon

mode are defined as

ak =1√2

(√

mνk~Qk − i

√

1

mνk~Pk

)

; a†k =1√2

(√

mνk~Qk + i

√

1

mνk~Pk

)

. (6.5)

I refer interested reader to Ref. [90] for further details of the derivations.

6.3 Universal Bosonic Simulation

The most general bosonic behaviour can be simulated if and only if the simulator can be

engineered to evolve under the most general Hamiltonian, which can be expanded as a

series of products of annihilation and creation operators as:

H(ai, a†i , t) =

∑

A(1)i (t)ai + A

(2,0)ij (t)aiaj + A

(1,1)ij (t)aia

†j + . . .+ h.c. . (6.6)

This Hamiltonian is time dependent, consists of superpositions of non-commuting terms,

and involves high order products of operators. Reproducing such a Hamiltonian requires

complicated engineering of the simulator, which is difficult in practice. Fortunately, Lloyd

and Braunstein [117] suggested three tricks to simplify the implementation of the general

evolution.

The first trick is to divide the evolution into short intervals, so the evolution operator

parameters, and treating the classical and quantum displacement, x(0)’s and q’s, as independent variables.

85


can be approximated as

U(t) ≈ exp(−iH(t1)δt) exp(−iH(t2)δt) . . . . (6.7)

At each short time interval δt, the evolution is determined by a time independent Hamil-

tonian at that time instance. Instead of engineering the time dependent Hamiltonian,

the evolution can be simulated by applying a sequence of time independent Hamiltonian.

The evolution simulated by this method is more accurate as δt shrinks. The aim of this

trick is to get rid of the time dependence of the Hamiltonian.

The second trick is to get rid of the superposition of non-commuting Hamiltonian.

This can be done by applying the Suzuki-Trotter expansion: for any Hamiltonian that

can be written as H = A+B, the evolution can be approximately constructed by applying

the components A and B in sequence, i.e.,

exp(−i(A + B)δt) ≈ exp(−iAδt) exp(−iBδt) +O(δt2) . (6.8)

This approximation is more accurate as δt shrinks.

The third trick is to implement highly nonlinear operation, i.e., evolution under the

Hamiltonian with higher than third orders of annihilation and creation operators, by

Hamiltonian with linear and less nonlinear operations. The key idea is that if two

Hamiltonians, A and B, are applied in appropriate sequence, the final evolution will

be determined by the commutator of A and B, i.e.,

exp(iAδt) exp(iBδt) exp(−iAδt) exp(−iBδt) = exp([A, B]δt2) +O(δt3) , (6.9)

where the higher order contribution, O(δt3), can be neglected when δt is small. Lloyd

and Braunstein [117] showed that if both A and B are higher than the third order, then

a Hamiltonian with arbitrary high order can be constructed by this method.

By using these three tricks, Lloyd and Braunstein showed that the evolution of arbi-

trary multi-mode Hamiltonian can be efficiently simulated by applying in sequence only

a basic set of operations. These operations include (i) displacement operator, which can

be implemented by applying a Hamiltonian with the first order of a and a†; (ii) squeezing

operator; (iii) phase-shift operator, which can be implemented by a Hamiltonian with

second order of a and a†; (iv) beam splitter, which can be implemented by a Hamiltonian

a†1a2+ h.c.; and (v) a nonlinear operator that is generated by higher than second order

terms of a and a†. The first four operators are regarded as the basic set of Gaussian

operations, which can transform a Gaussian Wigner function to any Gaussian Wigner

86


function (c.f. Sec. 3.2.2). They are also referred as the basic set of active linear elements

in quantum optics, which transforms an annihilation operator to any superposition of

annihilation and creation operators. The addition of the nonlinear operator, which is

also a non-Gaussian operator, is to break the Gaussianity of the operation 4.

6.4 Laser Implementation of Basic Operations

In this section, I present the strategies required to implement each of the five basic oper-

ators on the phonon modes of the ion trap bosonic simulator. The principle is to apply

laser field that couples the internal states and the motional states of the ion. Different

motional operations can be implemented by changing the frequency and magnitude of

the laser field.

Because the transition between internal state is undesirable in this situation, two laser

fields, where the frequency difference is much smaller than the internal state transition

frequency, is applied to induce Raman transition (see Fig. 6.2). Let us consider the

frequency, wavevector, and phase of the first (second) laser field are ω1 (ω2), ~k1 (~k2),

and φ1 (φ2) respectively. The effective frequency, effective wavevector, and phase of the

resultant Raman field are then ω = ω1 − ω2, ~k = ~k1 − ~k2, and φ = φ1 − φ2. As our

consideration is focused on the axial motion, the radial components of the Raman field

are neglected. This is possible because the radial direction is tightly trapped so that the

effective oscillation is off-resonant with the Raman field, and the angle between the laser

fields can be tuned so that ~k = kx~x is along the x axis. The Hamiltonian generated by

the Raman field is given by

VR = ~Ωeiφ exp (ikxq) e−iωt + h.c. , (6.10)

where Ω is the effective Rabi frequency. If the Rabi frequency of each laser field is Ω0,

then Ω = Ω20/∆, where ∆ is the detune of ω1 from ωgd

5.

Let us assume the Raman field is applied on the ith of the N ion in the chain, while

other ions are unaffected 6. The position operator in Eq. (6.10) is then belonging to the

ith ion, i.e., qi. As seen in Eq. (6.4), the motion of ith ion involves differently in the

collective motion of each mode, because of the inhomogeneity of eigenvector coefficients.

4This is analogous to the implementation of a discrete-variable UQC, which requires all the Cliffordgate (analogous to Gaussian operation) and one non-Clifford gate (analogous to non-Gaussian operation)

5I have defined the energy difference between internal state |m〉 and |n〉 to be ~ωmn.6Such a single ion addressing can be realised by using composite pulses [78].

87


(a) (b)

Figure 6.2: (a) Two laser fields with different frequency and wavevector are applied onan ion while other ions are assumed to be unaffected. Single ion addressing is possibleby using screening or composite pulses. (b) Internal level diagram of the ion. The laserfields interact with the ion through the dipole transition between the internal states |g〉and |d〉, but the laser frequencies are far-detuned from the transition frequency so thatno internal state is excited.

By inverting Eq. (6.4), the position operators can be expressed as

qi =N∑

k=1

βikQk , (6.11)

where the matrix βik is the inverse of the matrix αki. The coefficients of the matrices

can be found in Ref. [90]. By substituting Eq. (6.5) into Eq. (6.10), the Raman field

Hamiltonian can be written as

VR = ~Ωeiφ exp

(

i

N∑

k=1

ηik(ak + a†k)

)

e−iωt + h.c. , (6.12)

where the Lamb-Dicke parameters, ηik, are defined as

ηik = kxβik

√

~

2mνk. (6.13)

Typical ion trap experiments operate at the Lamb-Dicke regime, where η ≪ 1 7. The

Hamiltonian in Eq. (6.12) can be Taylor-expand in terms of η’s. In the interaction picture

with respect to the steady state Hamiltonian in Eq.(6.3), the Raman field Hamiltonian

7Motional and internal states can still be manipulated by laser field beyond the Lamb-Dicke regimeif the state-dependency of interaction coefficients is considered [177].

88


can be expressed as a series,

V(I)R ≈ ~Ωeiφ

(

I+ i

N∑

k=1

ηik(ake−i(νk+ω)t + a†ke

i(νk−ω)t)

+(i)2

2

N∑

k,k′=1

ηikηik′(akak′e−i(νk+νk′+ω)t + a†kak′e

i(νk−νk′−ω)t

+aka†k′e

−i(νk−νk′+ω)t + a†ka†k′e

i(νk+νk′−ω)t) + . . .)

+ h.c. . (6.14)

As we can see from this expression, the contribution of each term is determined by

an oscillating parameter, which the frequency depends on the Raman field frequency

and the mode frequencies. Each of the term can be made dominant by tuning the

Raman field frequency to be resonant, i.e., the oscillating term becomes a constant. The

contribution of other terms in the same order can be neglected according to rotating

wave approximation (RWA), while higher order terms are neglected because they scale

as higher power of small Lamb-Dicke parameters, i.e., Lamb-Dicke approximation (LDA).

In the following subsections, I will specifically discuss the implementation of each of the

basic operation. The validity of the approximations will be discussed later.

6.4.1 Displacement Operator

A displacement operator, D(α) = exp(αa† − α∗a), transforms the annihilation operator

as

a→ D†(α)aD(α) = a + α , (6.15)

where√

2~mν

Re(α) is the displacement of the collective position of the mode;√2~mνIm(α)

is the displacement of the collective momentum of the mode.

A displacement operator of the kth mode can be implemented by applying a Hamil-

tonian with the first order of ak and a†k [127, 134]. By applying a Raman field with the

frequency νk to the ith ion (assume the Lamb-Dicke parameter is nonzero for the mode),

the dominating terms in Eq. (6.14) will be

V(I)R ≈ i~ηikΩ(e

iφa†k − e−iφak) . (6.16)

After time t, the kth mode will be transformed as Eq. (6.15), where the displacement is

given by α = ηikΩeiφt. The value of the displacement is tuneable by changing the phase

and duration of the Raman field, as well as the effective Rabi frequency.

89


6.4.2 Phase-Shift Operator

A phase-shift operator P(θ) = exp(−iθa†a) transforms the annihilation operators as

a→ P†(θ)aP(θ) = ae−iθ , (6.17)

where θ is the phase shift.

A phase-shift operator of the kth mode can be implemented by applying a Hamiltonian

with the dominant term a†kak. This Hamiltonian can be realised by applying a Raman

field with ω = 0, i.e., both laser fields have the same frequency, to the ith ion. Then

Eq. (6.14) will become

V(I)R ≈ −~η2ikΩ(a

†kak + aka

†k) . (6.18)

After time t, the kth mode will be transformed as Eq. (6.17), where the phase shift

is given by θ = −2η2ikΩt. The value of the displacement is tuneable by changing the

duration of the Raman field and the effective Rabi frequency.

6.4.3 Squeezing Operator

A squeezing operator S(g) = exp(

(g∗a2 − ga†2)/2)

transforms the annihilation operator

as

a→ S†(g)aS(g) = cosh |g|a− g

|g| sinh |g|a† , (6.19)

where g is the complex squeezing parameter.

A squeezing operator can be implemented by applying a Hamiltonian that involves

second order terms of the annihilation and the creation operators, i.e. a2 and a†2. A

squeezing operator on the kth mode can be realised by applying a Raman field with

frequency 2νk to the ith ion. Then Eq. (6.14) will become

V(I)R ≈ −~

η2ikΩ

2(eiφa†2k + e−iφa2k) . (6.20)

After time t, the kth mode will be transformed as Eq. (6.19), where the squeezing pa-

rameter is g = −iη2ikeiφΩt, which is tuneable by changing the duration, the phase, and

the effective Rabi frequency of the Raman field.

6.4.4 Nonlinear Operator

Nonlinear operators transform an annihilation operator to an operator involving quadratic

and higher order terms of a and a†. It can be achieved by applying a Hamiltonian that is

90


at least the third order of a and a† [117]. For example, when a Raman field with ω = 3νk

is applied, the Hamiltonian in Eq. (6.14) will be dominated by

V(I)R ≈ −i~η

3ikΩ

6(eiφa†3k − e−iφa3k) . (6.21)

6.4.5 Beam Splitter

A beam splitter transforms the annihilation operators of two modes to be a superposition

of each other, i.e.,

a1 → cos θa1 + sin θeiφ′

a2 ; a2 → − sin θe−iφ′

a1 + cos θa2 , (6.22)

where θ is some real number. Such an operation can be implemented by an operator

B = exp(θ(eiφ′a†1a2 − e−iφ′

a†2a1)) .

The realisation of a beam splitter in trapped ion systems is first proposed in Ref. [180].

The proposed setup contains a single ion being trapped in an inhomogeneous three di-

mensional harmonic trap, so that the phonon mode frequencies are different in x, y, and z

direction. A beam splitter between two of the modes can be realised by applying Raman

radiation field with the frequency equal to the difference of the mode frequencies.

Here I extend this idea to a linear ion chain. By applying a Raman field to the ith

ion with frequency ω = νk−νl, the effective Hamiltonian in Eq. (6.14) will be dominated

by

V(I)R ≈ −~ηikηilΩ

2(eiφa†kal + e−iφa†l ak) . (6.23)

After applying the field for time t, the beam splitter operation will be implemented with

θ = ηikηilΩt/2 and eiφ′

= ieiφ.

The quality of this operation is determined by the validity of LDA and RWA, in other

words how dominating the terms are in Eq. (6.23) when comparing to other terms in

Eq. (6.14). More explicitly, the zeroth order term in Eq. (6.14) does not alter the per-

formance of beam splitter because it only contributes to a global phase. The first order

terms will exert a displacement operation on each of the mode, but the coefficient is mul-

tiplied by an off-resonant oscillation term. The variance of such erroneous displacement

scale as |ηimΩ/δ|2, where the detuning δ = |ω−νm| for the mth mode. For a perfect beam

splitter operation that the Raman pulse is exactly resonant with the frequency difference

of the kth and lth mode, the most seriously affected mode will be the one, say mth mode,

that νm is the closest to νk − νl. I have plotted the minimum value of |(νk − νl) − νm|among all k, l,m in Fig. 6.3(a). If the variance of the erroneous displacement has to be

91


2 4 6 8 100.0

0.1

0.2

0.3

Number of ions

Min

imum

|(ν

v −

ν u) −

νk|

(a)

2 4 6 8 100.00

0.02

0.04

0.06

0.08

Number of ions

Min

imum

|(

ν u−ν

v)−(

ν k−ν

l))|

(b)

Figure 6.3: (a) The minimum possible value of (a) |(νk−νl)−νm|, (b) |(νk−νl)−(νu−νv)|,for different number of ions in the linear trap. All the above values of frequencies are inthe unit of ν0.

less than 1%, the following criterion is implied:

ηmΩ . 0.1×min|(νk − νl)− νm| . (6.24)

To implement a beam splitter, the Hamiltonian in Eq. (6.23) is required to switch on

for about t ≈ 1/η2Ω, where the ηik’s are assumed to have the same order of magnitude.

In practice, t is desired to be shorter, preferably at the range of 100 µs 8. With the

criterion (6.24), the value of η can be estimated as

η & 105/min|(νu − νv)− νk| . (6.25)

If the number of modes to be simulates is less than 7, Fig. 6.3(a) shows that the minimum

value of |(νu−νv)−νk| is one order smaller than the trap frequency ν0. For state-of-the-art

value of ν0 = 2π × 106 MHz, η should be about 0.1.

Fig. 6.4 shows the numerical simulation of the evolution of state |1, 0〉 and |0, 1〉,where the first (second) index denotes the Fock state of centre-of-mass (stretching) mode9 of a two ion system. I vary the parameter η, where η =

√2η1 =

√2η2, and select the

appropriate Ω to retain the same evolution time. As the above estimation suggests, a

smaller η causes more serious mode population fluctuation due to the first order term

in Eq. (6.14). Therefore an appropriately large η should be chosen to implement a high

8Heating rate of trapped ion is as low as 70 quanta per second [36]. To suppress the error to lessthan 1%, i.e. heating rate is less than 0.01 quanta, the experiment has to be finished in about 100 µs.

9In centre-of mass mode, the ions always move in the same direction, i.e., in Eq. (6.4), α11 = α12 =1/

√2; in stretching mode, the ions always move in opposite direction, i.e., α21 = −α22 = 1/

√2.

92


0 20 40 60 80 100Time ( s)

0

0.2

0.4

0.6

0.8

1

Pro

bab

ilit

y

μ

Figure 6.4: Probability of |1, 0〉 (solid lines) and |0, 1〉 (dashed lines) after the beamsplitter Hamiltonian (Eq. (6.14) with ω = ν2 − ν1) is applied on the initial state |1, 0〉.The curves correspond to η = 0.2 (red), η = 0.1 (blue), η = 0.05 (black).

quality beam splitter operation.

If ηkΩ satisfies Eq. (6.24), increasing ηk can also speed up the beam splitter operation.

However, a large ηk also enhances the magnitude of the higher order terms in Eq. (6.14),

which would bring nonlinearity to the beam splitter. Particularly, the third order terms

is suppressed because it is scaled by the small Lamb-Dicke parameter, and it has to

be off-resonant. As discussed previously, the off-resonant effect can reduce a first order

term to be less significant than the second order resonant terms in Eq. (6.23), so the third

order term is expected to be less significant than the forth order term. For the forth order

terms, while most of them are also off-resonant, there are some, like a†kaka†kal, could be

in resonant with the terms in Eq. (6.23). These terms would produce a nonlinear phase

shift, which the magnitude scales as |η|2 and the phonon number of the involved mode,

i.e., 〈a†uau〉. In other words, the nonlinear error is more serious if more phonon is involved

in the quantum simulation.

For a simulation with a high phonon number, the η’s should be reduced. Therefore

the restriction in Eq. (6.25) has to be removed. Let us recall that Eq. (6.25) is imposed

because I require the first order term to be suppressed by the RWA. In fact, this restriction

can be relaxed if the first order terms can be suppressed by other methods. One method

is to add another Raman field which has the same phase and frequency but opposite

wave vector. The physical idea is to set up a standing wave and the ion locates at a node

of the radiation. Then the ions should not experience a state-dependent force. More

93


explicitly, the total Hamiltonian of the two Raman fields becomes

V(I)R2 = ~Ωeiφ

(

ei(kx−ωt) + ei(−kx−ωt))

+ h.c.

≈ ~Ωeiφ(

I+(i)2

2

N∑

k,k′=1

ηikηik′(akak′e−i(νk+νk′+ω)t + a†kak′e

i(νk−νk′−ω)t

+aka†k′e

−i(νk−νk′+ω)t + a†ka†k′e

i(νk+νk′−ω)t) + . . .)

+ h.c. . (6.26)

Because the first order terms cancel exactly, the condition in Eq. (6.24) is no longer

required to eliminate the first order term by off-resonance. Therefore, the value of ηΩ

can be increased for faster operation while η remains small for effective LDA.

Without the first order terms, the next major error source is the off-resonant second

order terms. These terms will induce beam splitting or two-mode squeezing operation

on other phonon modes. Suppose the uth and vth modes are affected, the variance of

the beam-splitting angle or the squeezing parameter oscillating scales as χ = |ηuηvΩ/δ|2,where δ = (νk − νl)− (νu± νv) is the detuning; the sign + (−) corresponds to the case of

erroneous squeezing (beam splitter). Therefore the beam splitter would negligibly affect

other modes if χ≪ 1 for any combination of k, l, u, v. If the smallest χ required is about

1%, the following criterion is imposed

0.1 min|(νk − νl)− (νu − νv))| & η2Ω , (6.27)

where all η’s are again assumed to be the same order of magnitude; and I have checked

that the contribution from the erroneous two-mode squeezing is less significant than the

erroneous beam splitter. Since the operation duration of a beam splitter is about 1/η2Ω,

Eq. (6.27) lower bounds the operation time. The minimum value of |(νk−νl)−(νu−νv))|is plotted in Fig. 6.3(b) for different number of ions. For example when N = 10, the time

required for a beam splitter on any two modes is at least 150 µs.

As a summary, a beam splitter can be implemented by applying a Raman field with

frequency tuned to the mode frequency difference of two modes. The operation time

of such approach is limited by Eq. (6.25) that guarantees the validity of RWA. Faster

operation can be implemented if an additional Raman field is applied, then the accuracy

criterion is improved to Fig. 6.3(b). I note that further speed up of beam splitter operation

may be possible if multiple pulses is applied, where erroneous Hamiltonian could be

eliminated by carefully choosing the pulse sequence [78].

94


6.5 Readout

For most of the investigation about bosonic systems, readout of boson number is particu-

larly important. In the current proposal of ion trap bosonic simulator, the phonon states

are measured by first coupling with electronic states and then conducting measurement

on the electronic states. Here I propose two ways to couple the phonon and electronic

states: by adiabatic passage and resonant flipping. I find that the adiabatic passage

approach can mimic the photon number non-resolving detectors in optical experiments,

but the operation has to be sufficiently slow to preserve the adiabaticity. On the other

hand the resonant flipping method can be much faster, but it can be applied with a priori

information about the phonon number.

6.5.1 Adiabatic passage

The idea of using adiabatic passage to create non-classical motional states of trapped

ions was first proposed by Cirac, Blatt and Zoller [49]. Inverting their protocol can map

the motional state to the electronic state of an ion. Let us consider a stimulated Raman

field with the effective frequency ω is interacting with the encoding states, |g〉 and |e〉,of the ion, the interaction potential is given by

V(I)T = ~Ωe−iωt(σ+e

iω0t+σ−e−iω0t)+i~Ωe−iωt

n∑

k=1

ηk(a†ke

iνkt+ake−iνkt)(σ+e

iω0t+σ−e−iω0t)+h.c. ,

(6.28)

where ω0 is the energy difference between the encoding states. If the kth mode is to be

measured in the Fock state basis, the ω is tuned to ω = ω0−νk+∆(t). ∆(t) is time varying

but the amplitude is always small when comparing to νk. The coupling of electronic

states to other phonon modes, as well as the carrier transition between the electronic

states, are off-resonant and so their contribution is suppressed. The Hamiltonian can be

approximated as involving only the dominating term, i.e.,

V(I)T = i~Ωηk(σ+ake

−i∆(t)t − σ−a†ke

i∆(t)t) . (6.29)

The energy levels and the Raman field frequencies in an adiabatic transfer are shown in

Fig. 6.5(a).

If ∆(t) is slowly varying, the system will remain in the same level throughout the

whole process. The energy states change adibatically as follows,

|g 0〉 → |g 0〉 ; |g n〉 → |e n− 1〉 ; |e n〉 → |g n+ 1〉 , (6.30)

95


ν

(a) (b)

ν

Figure 6.5: Energy levels of an ion during measurement processes. The dashed line is avirtual energy level that is detuned from an electronic state. The dotted lines are sidebandenergy levels. (a) Energy levels during adiabatic transfer. The Raman field frequencyis tuned from slightly smaller than the red sideband (light grey) to slightly higher thanthe red sideband (dark grey). (b) Raman field frequencies of the red sideband transition(red arrows) and blue sideband (blue arrows). That of the carrier transition is shown inFig. 4.2.

where the second index in the bra denotes the phonon number state.

A phonon number non-resolving detector can be implemented by separating the |0〉from other phonon number state. I assume all ions are initially in |g〉. Let us consider ageneral motional state of the kth mode is |ψ〉 =∑∞

m=0 αm|m〉. When adiabatic passage

is applied, the total state transforms as

|g〉(α0|0〉+ α1|1〉+ α2|2〉 . . .) → α0|g 0〉+ |e〉(α1|0〉+ α2|1〉+ . . .) . (6.31)

The electronic state is then measured by standard fluorescence measurement techniques

discussed in Sec. 4.2.2. From Eq. (6.31), the measurement of |e〉 is equivalent to the

measurement of nonzero phonon state, which is an phonon analogous of a photon number

non-resolving detector.

Incorporating with carrier pulses (The pulse that transits electronic states |g〉 and

|e〉 while leaving motional states unchanged. More will be discussed in Sec. 6.5.2.) and

an additional meta-stable state |r〉, the method of adiabatic passage can be extended to

projectively measure any Fock state. More explicitly, after the operation in Eq. (6.31),

carrier transition and subsequently another round of adiabatic passage are applied. The

96


state will be transformed as

α0|g 0〉+ |e〉( ∞∑

n=0

αn+1|n〉)

→ α0|e 0〉+ |g〉( ∞∑

n=0

αn+1|n〉)

→ |g〉(α0|1〉+ α1|0〉) + |e〉( ∞∑

n=0

αn+2|n〉)

. (6.32)

Repeating this process for k times, all the components with phonon number n 6 k will be

coupled to |g〉, while those with n > k will be coupled to |e〉. Finally, adiabatic passage

is conducted to push all states with n < k to |r〉. The overall operation is

|g k〉 → |g 0〉 ; |g n〉 → |r k − n〉(for n < k) ; |g n>k〉 → |e n− k〉(for n > k) . (6.33)

Measuring the electronic state |g〉, |e〉, |r〉 corresponds to the measurement with the

projection-valued measurement (PVM)

|k〉〈k|, I<k, I>k , (6.34)

where I<k (I<k) is the identity operator of the subspace with phonon number n < k

(n > k). To the best of our knowledge, such a measurement scheme does not exist in

quantum optics experiments. The scheme would be a useful tool in analysing the boson

number population of a general motional state, and also in quantum information tasks

that involve post-selection of phonon number states.

The successful rate of adiabatic state transfer is higher for a larger Landau-Zener

parameter, i.e. Γ ≫ 1 (see e.g. [183] for reference), where

Γ =(ηkΩ)

2

∆(t)≫ 1 . (6.35)

Suppose the detuning varies linearly as time from −∆0 to ∆0, then ∆(t) = 2∆0/T for a

total transfer time T . A large Γ can be achieved by setting a long T or a small ∆0.

The magnitude of ∆0 has to be bounded to avoid mixing of energy eigenstates. Let

us first consider the lower bound. The Hamiltonian in Eq. (6.29) at t = 0 is equivalent

to the interaction picture of the following Hamiltonian

H = ~∆0|e〉〈e|+ i~Ωηk(σ+ak − σ−a†k) , (6.36)

of which the dressed energy eigenstates, expressed in terms of |g n〉 and |e n〉, are given

97


by

|En±〉 =√n + 1ηk~Ω

√

(En±)2 + (n+ 1)(ηkΩ)2|g n+ 1〉+ i

En±√

(En±)2 + (n + 1)(ηkΩ)2|e n〉 , (6.37)

where En± = 12(∆0 ±

√

∆20 + 4(ηkΩ)2) are the eigenenergies. Since adiabatic passage

transfer states from |En−〉 to |En+〉, it can be used to transfer |g n + 1〉 to |e n〉 if

|〈En−|g n+ 1〉|2 and |〈En+|e n〉|2 are close to 1. This implies the condition

∆0 ≫√nmaxηkΩ , (6.38)

where nmax is the maximum phonon number involved in the simulation. Combining with

the criterion of adibaticity, Eq. (6.35), the total transfer time should obey T ≫ 1/∆0.

On the other hand, ∆0 has to be much smaller than νk, in order to prevent the

mixing of |g n〉 and |e n〉 due to the carrier transition terms, i.e., ~Ω(σ− + σ+), that

are of the zeroth order of η (to be discussed in Sec. 6.5.2). A prudential choice is to

set ∆ ≈ ηkνk. The duration of an adiabatic state transfer can then be estimated by

using typical parameters in state-of-the-art ion-trap experiments, i.e., ν ≈ 106 MHz and

η ≈ 0.1. If each “≫” relationship is assumed to denote a difference of one order of

magnitude, then T ≈ 10−3 s.

Changing the time dependence of ∆(t) causes little effect on the required time T .

Alternatively, T can be suppressed by an order of magnitude if the Rabi frequency is also

time dependent. One of the scheme is proposed by Allen and Eberly in Ref. [10, 44],

which the Rabi frequency and detuning vary as

Ω(t) = Ω0sech

(

(t− T

2)1

t0

)

; ∆(t) = ∆0tanh

(

(t− T

2)1

t0

)

, (6.39)

where t0, Ω0 and ∆0 are respectively the characteristics time, Rabi frequency, and de-

tuning of the process. The successful transfer rate of using this scheme is higher than

the stardard linear-varying detuning, because the initial state mixing is small due to the

small initial Rabi frequency. Besides, ∆ varies slowly at the vicinity of ω ≈ ω0 − νk so

the Landau-Zener parameter is large that exhibits high adiabaticity. If Eq. (6.29) is the

exact expression for the Hamiltonian, the probability of the state transfer from |g n〉 to|e n− 1〉 will be given by [44]

P = 1− sech2(π

2∆0t0

)

cos2(

πt02

√

nη2Ω20 −∆2

0

)

. (6.40)

98


The cos term is bounded by 1 when nη2Ω20 > ∆2

0, therefore a high transfer rate is

determined by a small sech term, i.e. a large ∆0t0. On the other hand, the cos term

will become cosh, which is a diverging function, when nη2Ω20 < ∆2

0. Therefore the Rabi

frequency has to be reasonably large, i.e., Ω0 & ∆0/ηnmax. However a too large Ω0 will

break the RWA of the zeroth order term in Eq. (6.28). Therefore the parameters Ω0, ∆0,

and t0 have to be optimised for a fast and accurate adiabatic transfer. For a bosonic

simulator with two ions in a trap with ν = 2π MHz, an adiabatically transfer for nmax = 3

with 0.99 fidelity takes only 80 µs 10.

6.5.2 Resonant Pulses

When the Raman field frequency is tuned exactly to the electronic state transition fre-

quency, Eq. (6.28) will be dominated by the zeroth order term, viz.

V(I)T ≈ ~Ω(σ+ + σ−) . (6.41)

which will induce Rabi transition between the electronic states, by the motional state is

not affected if the RWA is effective. This operation is referred as the carrier transition.

The motional and electronic states of an ion are coupled by tuning the laser frequency

to be resonant to motional sideband frequencies. When the frequency is tuned to the red

sideband, i.e., ω = ω0 − νk, under RWA the Hamiltonian in Eq. (6.28) is dominated by

V(I)T ≈ i~Ωηk(σ+ak − σ−a

†k) . (6.42)

The evolution operator of this Hamiltonian can be reduced to a tensor product of trans-

formations within the subspaces |g n + 1〉, |e n〉, viz.

|g 0〉 → |g 0〉 (6.43a)

|g n+ 1〉 → cos(

Ωηk√n+ 1t

)

|g n+ 1〉+ sin(

Ωηk√n + 1t

)

|e n〉 (6.43b)

|e n〉 → − sin(

Ωηk√n + 1t

)

|g n + 1〉+ cos(

Ωηk√n + 1t

)

|e n〉 . (6.43c)

On the other hand, the blue side-band transition is activated when ω = ω0 + νk.

Under RWA, the Hamiltonian in Eq. (6.28) is dominated by

V = i~Ωηk(σ−ak − σ+a†k) . (6.44)

10The parameters for this transfer are η = 0.2, Ω0 = 2.25MHz, and ∆0 = 0.1ν

99


Under this Hamiltonian, the dressed states transform as

|e 0〉 → |e 0〉 (6.45a)

|g n〉 → cos(

Ωηk√n + 1t

)

|g n〉 + sin(

Ωηk√n+ 1t

)

|e n+ 1〉 (6.45b)

|e n+ 1〉 → − sin(

Ωηk√n+ 1t

)

|g n〉+ cos(

Ωηk√n+ 1t

)

|e n+ 1〉 . (6.45c)

The Raman field frequencies of the resonant pulses are shown in Fig. 6.5(b)

The major difference between the resonant-pulse method and adiabatic passage is

that the transition amplitude depends on the phonon number. As an example, let us

consider the state that is a superposition of |0〉, |1〉, |2〉. When |g 1〉 is transferred to |e 0〉by a red sideband pulse with t = π/(2Ωηk), other photon states are transited as

|g 0〉 → |g 0〉 ; |g 1〉 → |e 0〉 ; |g 2〉 → cos

(√2π

2

)

|g 2〉+ sin

(√2π

2

)

|e 1〉 . (6.46)

Since |2〉 will be coupled to both |g〉 and |e〉 after the pulse, afterwards measurement

on the electronic state cannot provide precise information about the phonon number

of the initial state. If the maximum phonon number in the state is known, in this case

nmax = 2, the ground phonon state could still be singled out to |g〉 by applying a sequence

of sideband different phases [78]. In general if no a priori information about the phonon

state is known, singling out a Fock state is non-trivial. In the following, I will describe

two tricks to apply the resonant pulse to conduct projective measurement on particular

a phonon state |k〉.

For demonstrating the ideas in future discussions, I introduce the circuit diagram in

Fig. 6.6.

6.5.2.1 Post-selection Method

The first trick is to use a sequence of measurement to remove unwanted superpositions,

and then post-select the desired outcomes. Let us consider if |0〉 is to be post-selected

from a state |ψ〉 = α0|0〉 + α1|1〉 + α2|2〉. After applying a red sideband pulse with

t = π/(2Ωηk), a fluorescence measurement on |e〉 state is conducted. Depending on the

100


Bg,e

(a) Blue sideband transition cou-pling with |g n〉 ↔ |e n + 1〉transition. The operation followsEq. (6.45) with parameter θ = Ωηktand φ depends on phase of the Ra-man field.

Rg,e

(b) Red sideband transition cou-pling with |g n〉 ↔ |e n − 1〉transition. The operation followsEq. (6.43) with parameter θ = Ωηktand φ depends on phase of the Ra-man field.

g,e

(c) Carrier transition withangle (θ, φ) between |g〉, |e〉.The operation followsEq. (6.41) with parameterθ = Ωt and φ depends onphase of the Raman field.

(d) Evolution of the elec-tronic state (solid line) andthe phonon state (dashedline).

BS BSg

(e) Beam splitter generated by Raman field cou-pling to |g〉. The operation follows Eq. (6.22)with angle θ = ηuηvΩt and φ depends on phaseof the Raman field.

g

(f) Fluorescencemeasurement on theelectronic state |g〉.

Figure 6.6: List of elements in circuit diagram of boson simulation.

measurement outcome, the state becomes

α0|0〉+ cos

(√2π

2

)

α2|2〉 (electronic state is not |e〉) ; (6.47)

α1|1〉+ sin

(√2π

2

)

α2|2〉 (electronic state is |e〉) , (6.48)

up to some normalization constant. If the electronic state is |e〉, the ion will recoil when

scattering the measurement pulse; the gain in momentum will add significant noise to

the motional state. In other words, the phonon information is lost if positive outcome

is obtained and the simulation has to be terminated. On the other hand, the motional

state is insignificantly affected if the electronic state is not in |e〉, the simulation can then

be further proceeded. The aim of this step is to remove the amplitude of |1〉 from the

superposition. The next step is then to remove |2〉 by a transition pulse with duration

π/(2√2Ωηk) that flips |g 2〉 to |e 1〉. A negative result of |e〉 measurement is then a

projective measurement on |0〉. The procedure is outlined as circuit diagram in Fig. 6.7.

The above scheme can be extended to measuring |0〉 from a wider range of phonon

state. The most direct method is to remove the amplitude of |n〉 in an ascending order

of n, but the number of measurement is equal to nmax. Alternatively, let us consider that

each round of resonant transition and measurement post-selection does not only remove

101


Rg,e

eclick

Rg,e

eclick

Figure 6.7: Circuit diagram of phonon non-distinguishing measurement using the post-selection method. If all electronic state measurements are negative, the initial motionalstate is projected onto the ground state.

the superposition of a particular |m〉, but also weakens the amplitude of other states

by a factor of cos (√mπ/2

√n). The cumulative effect of the suppression factors are not

evenly distributed; some Fock states remain prominent after a few rounds of operation.

A shortcut of measurement sequence is to remove those prominent states instead of

following the ascending order of phonon number. Let us consider if the PVM |g 0〉〈g 0|is to be imposed on a state with nmax = 30. Let the state after post-selection to be

|ψ〉 → O|ψ〉. For a high fidelity measurement, the post-selection process would map any

state to be close to |g 0〉. The fidelity of this measurement method can be defined as

F = 〈g 0|O(|g〉 ⊗ I)(〈g| ⊗ I)O†|g 0〉 , (6.49)

where the averaged input state is given by I, which is the identity operator of the subspace

with n ≤ 30. If six rounds of measuring sequence is to remove the state |1〉 to |6〉, then the

measurement fidelity is F = 0.62. On the other hand, if the six rounds of measurement is

to remove |1〉, |2〉, |3〉, |4〉, |7〉, |13〉, then the measurement fidelity is improved to F = 0.99.

In Fig. 6.8, I show the probability of remaining in the state after a measurement sequence,

if the initial state is a Fock state.

Lastly, the above method can be further generalised to measure any Fock state |m〉.Any Fock state can be transferred to the ground state by applying alternative red side-

102


Figure 6.8: Probability of the first 30 Fock states in the post-selected branch after re-moving the |1〉 (hollow square), |1〉 and |2〉 (hollow circle), |1〉, |2〉, |3〉 (filled triangle),|1〉, |2〉, |3〉, |4〉, |5〉, |6〉 (filled diamond), |1〉, |2〉, |3〉, |4〉, |7〉, |13〉(filled square, very closeto zero).

band and carrier pulses, i.e.

|g m〉 red−−−→ |e m− 1〉 carrier−−−−−→ |g m− 1〉 . . . |g 0〉 . (6.50)

Other Fock state components will be transformed to some superpositions of |g n + 1〉and |e n〉, except |g 0〉. |e〉 is then measured and the state is retained if negative result is

obtained, in order to remove the phonon states associated with |e〉. Conducting the abovepost-selection measurement scheme is then equivalent to measuring the PVM |m〉〈m| onthe original state. Following the same principle, a PVM of any motional state, |ψ〉〈ψ|,can be measured, because an arbitrary motional state can be mapped to the ground state

by a sequence of resonant pulses [69].

I note that this post-selection method preserves the total probability of measuring

any state |m〉, in other words the accumulative probability of measuring |m〉 from |ψ〉 =∑

n=0 αn|n〉 remains |αm|2. Furthermore, if this phonon mode is entangled with other

modes as |Ψ〉 =∑

n=0 αn|n〉|φn〉, the measurement sequence preserves the associated

component in other degree of freedom, i.e., |φm〉. The above measurement scheme is

useful to investigate quantum information protocols that involves entanglement and post-

selection, such as the linear optics quantum logic gate [101].

6.5.2.2 Multiple Electronic State Method

A more powerful phonon number-resolving measurement can be achieved by using multi-

ple electronic states. Although the number of phonon state can be measured is restricted

103


by the number of meta-stable state available in an ion, this method is robust and achiev-

able by current technology, Besides, it is particularly useful for small scale bosonic sim-

ulation, such as the demonstration of the Hong-Ou-Mandel (HOM) effect that will be

discussed in the next section.

As an illustration, let us consider three electronic states |g〉, |e〉, and |r〉 are availablefor the laser manipulation. If a motional state with nmax = 2 is to be measured, a

red sideband pulse of the |g〉 ↔ |e〉 transition is first applied with the duration t =

π/(2√2Ωηk). The Fock state components will transform as

|g 0〉 → |g 0〉 ; |g 1〉 → cos

(

π

2√2

)

|g 1〉+ sin

(

π

2√2

)

|e 0〉 ; |g 2〉 → |e 1〉 . (6.51)

Then a red sideband pulse of the |e〉 ↔ |r〉 transition with duration t = π/(2Ωηk) is

applied and transforms the states as

|g 0〉 → |g 0〉 ;

cos

(

π

2√2

)

|g 1〉+ sin

(

π

2√2

)

|e 0〉 → cos

(

π

2√2

)

|g 1〉+ sin

(

π

2√2

)

|e 0〉 ;

|e 1〉 → |r 0〉 . (6.52)

Finally, a red sideband pulse of the |g〉 ↔ |r〉 transition with duration t = (√2 −

1)π/(2√2Ωηk) is applied, which transforms

|g 0〉 → |g 0〉 ; cos

(

π

2√2

)

|g 1〉+ sin

(

π

2√2

)

|e 0〉 → |e 0〉 ; |r 0〉 → |r 0〉 . (6.53)

After the above operations, each phonon state, |0〉, |1〉, |2〉, is then associated to an

electronic state, |g〉, |e〉, |r〉 respectively. Therefore the measurement of electronic states

is equivalent to a phonon number resolving detection. The procedure is outlined as a

circuit diagram in Fig. 6.9. I note that the above sequence is applicable only if nmax = 2.

For a system with a larger nmax, components of higher number Fock states will distribute

on the three electronic states, so a dedicated pulse sequence has to be designed for each

nmax.

6.6 Initialization

Before the simulation, all the ions are prepared in the ground electronic state, |g〉, byoptical pumping, and the collective modes are cooled to the ground motional state, |0〉,

104


R

R

g,e

e,r

Rg,e

g,e,r

Figure 6.9: Circuit diagram of phonon number resolving measurement for nmax = 2 withthree electronic states involved.

by laser cooling [181, 78]. As I have shown that the universal set of bosonic operators can

be implemented in the UBS, arbitrary phonon state |ψ〉 can be initialised by applying

a unitary transformation that maps U |0〉 = |ψ〉. In Bosonic simulation, there are input

states that are of particular interest. These include Gaussian states, such as coherent

states, single mode squeezed states, and multimode squeezed states, and non-Gaussian

states, such as Fock states and Schrodinger Cat state.

For Gaussian states, coherent states can be initialised by applying the displacement

operator in Sec. 6.4.1 to the ground motional state. Similarly, squeezed vacuum state

can be initialised by applying the squeezing operator in Sec. 6.4.3 to the ground motional

state. A multimode squeezed state can be realised by applying in beam splitter Sec. 6.4.5

to single mode squeezed states [176].

Fock states can be initialised by applying alternative sideband transitions. More

explicitly, a blue sideband pulse with duration t = π/2ηΩ transforms |g 0〉 to |e 1〉. Thena red sideband transition with duration t = π/2

√2ηΩ transforms |g 1〉 to |e 2〉. This

cycle of sideband pulses retain the electronic state as |g〉 while a Fock state with n = 2

is created. Higher number Fock states can be created by continuing this sequence of

blue and red sideband transition (with appropriate pulse duration, which depends on the

initial phonon number state in that round). I note that if an odd number phonon state

is to be created, one of the sideband pulses can be replaced by a carrier pulse. A Fock

state with n = 2 has been realised in ion trap experiments [127].

Schrodinger Cat states can be created by applying displacement operators and carrier

105


pulses [134]. Firstly, a carrier pulse is applied to create a superposition of electronic

state, i.e., (|g〉+ |e〉)/√2. Then a displacement operator in effective with |g〉 is applied to

transform the total state as (|g〉|α〉+ |e〉|0〉)/√2, where |α〉 is a coherent state of motion.

Similarly, a displacement operator in effective with |e〉 is applied to transform the total

state as (|g〉|α〉+ |e〉|−α〉)/√2. Finally, a carrier pulse is applied to create superpositions

of electronic states, the total state is then given by |g〉(|α〉+ |−α〉)/2+ |e〉(|α〉−|−α〉)/2.Obviously, the motional states associated to each of the electronic state are Schrodinger

Cat states with different phases. Subsequent simulation operations can be applied in

effective with |g〉. At the end of the simulation, the electronic state of the ion is measured.

By post-selecting the case that |g〉 is measured, the result of the simulation on |α〉+ |−α〉can be obtained.

I note that although both displacement operators are implemented by two laser fields

with the frequency difference ν, the operation is electronic state dependent. Let us

consider the operator in effective with |e〉. The laser fields are roughly ∆ detuned from

the transition between |e〉 and |d〉, which is the intermediate state that facilitates Raman

transition (c.f. Fig. 6.2(b)). The field frequencies are detuned from the transition between

|g〉 and |d〉 by ∆+ωe−ωg, which is much larger than ∆. Therefore the effective coupling

to |g〉 is small, and hence the motional state associated to |g〉 is barely affected. The

situation is similar for the displacement operator in effective with |g〉.In the context of Gaussian state initialisation, the ion trap UBS described in this

chapter offers no advantage over optical experiments, because the creation of Gaussian

photon states has been realised with high fidelity. On the other hand, due to the weak

nonlinear interaction of photons, non-Gaussian optical states have to be created by in-

efficient techniques, such as post-selection and using nonlinear optical elements. How-

ever, Fock states and Schrodinger Cat states have been efficiently created in ion trap

experiments [127, 134]. Therefore, realising an ion trap UBS allows us to study how non-

Gaussian states evolve under general interaction, which is difficult to observe in optical

experiments.

In Appendix A.5, I outline the procedure of demonstrating the HOM effect.

106

Chapter 7

Ion Trap Bosonic Simulator 2: Ions

in Separate Trap

7.1 Introduction

Although the number of ions in a trap is not fundamentally limited, the bosonic simulator

architecture presented in Sec. 6 is not very scalable. This is because the running time of

a bosonic simulation increases as more ions and modes involved, so the maximum scale of

a bosonic simulation is limited by the heating and the decoherence rate of the motional

states. The reason is that the addition of ions will narrow the frequency gap between

phonon modes, so sideband transitions have to be conducted slowly to avoid significant

errors. In addition, a measurement on one phonon mode via resonance fluorescence

may cause significant heating of the ion chain, which distorts the states of other phonon

modes. These shortcomings limit the number of modes and the population of phonons

that can be simulated accurately.

The problem of an excess of ions in a single trap also appears in ion trap quantum

computing [78]. As discussed in Sec. 4.3, these problems can be solved by adopting the

KMW architecture: chains of small number of ion qubits are stored in separate locations

of an array of traps, so that the manipulation on one qubit negligibly affects the others,

and the quality of individual logic operation is independent of the scale of quantum

computation. Considerable advances in experimental realization of these ideas have been

made in the past few years [150, 30, 33, 175]; in particular, entanglement gates have been

performed on ions which were initially far-separated, and ions have been moved between

traps.

In this chapter, I propose to use the KMW idea to implement a UBS on trapped ions

107

Chapter 7. Ion Trap Bosonic Simulator 2: Ions in Separate Trap

system. I consider each ion to be stored in a separate harmonic trap, in which only one

bosonic mode is present. All single mode operations can be conducted by either changing

the storage trap potential or by laser manipulations. The linear beam splitter is based on

Coulomb interaction, which is the same principle as the kinetic energy exchange in Refs.

[36, 79]; the difference here is that the distance between ions is variable in order to speed

up the process. Ions can be transported in specific trajectories that do not cause motional

excitations [168, 46]. The advantages of the current scheme are that the quality of each

operation is independent of the number of modes involved in the simulation, and the

initialization and readout of any one mode will not distort the others. This architecture

of bosonic simulator is thus more scalable.

I begin by presenting the setup and the physical model of the proposal in Sec. 7.2.

The implementation of single mode operations is introduced in Sec. 7.3. In Sec. 7.4, I

show that a linear beam splitter can be implemented by precisely combining and splitting

two traps through changing the quadratic and quartic potentials. The initialization and

readout of phonon states are presented in Sec. 7.5. This chapter is summarized in Sec.

7.6 with some discussions. Most of the material of this chapter is included in a paper

published by Daniel James and myself [107].

7.2 Model

I again assume ions are tightly trapped in the y and the z directions by a strong rf field

while a weaker dc potential is applied along the x direction. I assume this configura-

tion would effectively restrict the ion to move along only the x direction because the

excitations in other directions are negligible.

The configuration of the system is schematically shown in Fig. 7.1. Ions are trapped in

an array of harmonic storage traps, and only one ion is present in each trap. The distance

between the equilibrium position of two neighbouring traps is L, which is sufficiently large

that Coulomb coupling between the ions can be neglected. Hence the total Hamiltonian

of the system is given by

H0 =∑

n

p2n2m

+1

2mν20 x

2n , (7.1)

where the subscript n denotes the quantities belonging to the ion in the nth trap; xn is

the position operator of the nth ion with respect to the equilibrium position of the nth

trap, i.e., position operator of the quantum fluctuation; pn is the momentum operator of

108


Figure 7.1: The configuration of the current ion trap UBS is an array of storage traps.Only one ion is trapped in the axial harmonic potential in each trap. The traps areseparated by a distance L, which is large enough to prevent disruption from the others.The position displacement of the ith ion, xi, is accounted with respect to the centre ofthe ith trap.

the nth ion. The annihilation operator of the phonon mode of the nth ion is given by

an =

√

mν02~

xn + i

√

1

2~mν0pn . (7.2)

The ions are cooled to both the motional and electronic ground state before the

simulation. The trap potentials will be varied, but the potentials should return to the

original form in Eq. (7.1) after each operation. Each operation is characterized by the

transformation of the motional state in the interaction picture. By generalising the

discussion in Sec. 4.3.2, a multiple ion state is transformed in the interaction picture as

|ψI(T )〉 = S|ψI(0)〉 , (7.3)

where S ≡ exp(iH0t/~)U is the S-matrix; U is the evolution operator in the Schrodinger

picture; H0 is the multiple ion storage Hamiltonian in Eq. (7.1). The annihilation oper-

ator of each mode in the interaction picture is then transformed as

an → S†anS = U †S anUSe

iω0T . (7.4)

I will omit the n in future discussions of single mode operation.

7.3 Single Mode Operations

As discussed in Sec. 6.3, any single mode operator can be achieved by alternatively

applying the displacement operators, phase-shift operators, squeezing operators, and a

109


(a) (b) (c)

Figure 7.2: (a) A displacement operator is implemented by changing the trap centre ofthe harmonic well. (b) A phase-shift operator or a squeezing operator is implementedby varying the harmonic potential strength. (c) An extra quartic potential is applied toimplement the nonlinear phase gate.

nonlinear operator [117, 176]. In Sec. 6.4, I have discussed that each of these operators

could be implemented by applying laser fields with different frequencies. However, the

accuracy and speed of these laser-mediated operations are limited by the validity of the

LDA and RWA [180, 112]. In this section, I consider an alternative approach that the

operators are implemented by varying the harmonic trap potential or by perturbatively

applying a quartic potential. Both the harmonic and the quartic potential can be realised

in experiments [85]. In future discussions, I assume the operations are operating from

t = 0 to t = T . A summary of the operations is shown in Fig. 7.2.

7.3.1 Displacement Operator

Apart from the laser-mediated method presented in Sec. 6.4.1, another way to perform

the displacement operator is to move the harmonic trap, i.e. replacing the static storage

trap by a displaced harmonic well with constant trap strength. The Hamiltonian of such

potential is the same as Eq. (5.1), i.e.,

HD =p2

2m+

1

2mν20 (x− s(t))2 ; (7.5)

where s(t) specifies the path of the trap centre. According to the discussion in Sec. 4.3.1

and Sec. 5.2, this potential implements a displacement operator on the motional states.

I require the trap centre locates at the origin before and after the operation, i.e.

s(0) = s(T ) = 0. Then every coherent state, |χ〉, will be transformed as

|χ〉 →∣

∣

∣

(

χ−√

mν02~

∫ T

0

s(t) exp(iν0t)dt

)

e−iν0T⟩

(7.6)

up to a global phase that will not affect the simulation result [106]. In the interaction

picture, the annihilation operator transforms according to Eq. (7.4) is

a→ a−√

mν02~

∫ T

0

s(t) exp(iν0t)dt ≡ a+ α[s(t)] , (7.7)

110


where the displacement α is a functional of s(t).

The s(t) that produces a specific displacement α0 is not unique. The appropriate s(t)

can be obtained systematically by the inverse engineering method described in Ref. [168],

or by a simpler method that bases on the linearity of the displacements and paths. The

later method is introduced as follows. First of all, we guess two arbitrary paths, s1(t)

and s2(t), that satisfy the boundary conditions s(0) = s(T ) = 0. According to Eq. (7.7),

the paths will produce two displacements, α1 ≡ α[s1] and α2 ≡ α[s2]. α1 and α2 should

not be scaled by a real number, otherwise another path s3(t) has to be guessed. If the

requirement is satisfied, there must exist two real parameters, γ1 and γ2, such that

α0 = γ1α1 + γ2α2 . (7.8)

Due to the linearity of the functional α, the path s(t) = γ1s1(t) + γ2s2(t) will give the

desired displacement α0.

7.3.2 Squeezing Operator

As discussed in Sec. 6.4.3, a squeezing operator can be realised by applying a Raman

field with ω = 2ν0 to the ion. However, the operation time of this method should be

much longer than 1/ν0 for RWA to be valid. I here describe an alternative approach to

realise a squeezing operator by varying the trap potential. Let us consider the storage

potential is replaced by a harmonic well with varying trap strength. The Hamiltonian of

such potential is given by

HS =p2

2m+

1

2mν2(t)x2 , (7.9)

The trap frequency is required to return to that of the storage trap after the operation,

i.e. ν(0) = ν(T ) = ν0. According to Eq. (4.40), the operation transforms the annihilation

operator as

a→ η∗(T )e−i(Θ(T )−ν0T )a− ζ(T )ei(Θ(T )+ν0T )a† , (7.10)

where there is no displacement because the harmonic well centre is fixed. The above

operation coincides with a squeezing operation if |ζ(T )| 6= 0.

By comparing Eqs. (7.10) and (6.19), the magnitude of the squeezing parameter is

related to the auxiliary function as |η| = cosh |g|, and the phase of the squeezing operator

is given by g|g| = |η(T )

ζ(T )| ζ(T )η∗(T )

e−2i(Θ(T )−ν0T ). I note that the evolution of the phonon mode

is exactly described by the analytical solution of the time dependent harmonic oscillator,

so the duration of the above squeezing operation is not limited by the validity condition

of the RWA.

111


Here I describe a systematic process to deduce a ν(t) for implementing any squeezing

parameter, g. By putting the general solution of b(t) in Eq. (4.22) into Eq. (4.28), it can

be checked that |η(t)| = cosh(δ/2), and hence δ = 2|g|. Therefore, obtaining a desired

magnitude of the squeezing parameter is equivalent to obtaining a ν(t) that b(t) acquires

a desired δ after the operation.

Such a condition can be satisfied by a wide range of ν(t); a particular ν(t) can be

obtained inversely from an ansatz of b(t). An example is

bS(t) =√

cosh δ + sinh δ sin(2ν0t)h(t) + (1− h(t)) , (7.11)

where h(0) = 0 and h(T ) = 1. ν(t) should be continuous before and after the operation,

so bS(t), bS(t), and bS(t) have to be continuous at t = 0 and t = T . For instance,

h(t) = 10(t/T )3 − 15(t/T )4 + 6(t/T )6 meets the requirement. The time variation of ν(t)

can then be obtained by inputting bS(t) into Eq. (4.14).

I note that the above method may not construct the correct phase of the desired

squeezing operator, but the phase can be rectified by applying phase-shift operator after

the squeezing operation.

7.3.3 Phase-Shift Operator

As discussed in Sec. 6.4.2, a phase-shift operator can be realised by applying a Raman

field to the ion with ω = 0. The drawback of this method is that in order for the a†a

term to be dominant, the operation time has to be much longer than 1/ν0 for an effective

RWA. I here describe an alternative implementation of a phase-shift operator that the

operation time can be reduced to the same order as 1/ν0.

The method is similar to the squeezing operation described in Sec. 7.3.2, i.e., by

varying the harmonic potential of the trap. The only difference here is the boundary

conditions are set as η(t ≥ T ) = 1 and ζ(t ≥ T ) = 0, in order to avoid parametric

excitation. This criterion is equivalent to require the auxiliary function, bφ(t), to satisfy

the boundary conditions

bφ(t ≤ 0) = 1 ; bφ(t ≥ T ) = 1 . (7.12)

According to Eq. (4.40), such a harmonic potential variation will transform the annihi-

lation operator as

a→ ae−i(Θ(T )−ν0T ) , (7.13)

where there is no displacement because the harmonic trap centre is fixed; Θ(T ) is de-

112


pending on the auxiliary function via Eq. (4.21). Obviously, the operation realises a

phase-shift operator.

The above conditions do not impose a unique form of ν(t) and bφ(t), therefore we have

the freedom to choose ν(t) in a manner that is convenient in practice. Alternatively, we

can guess an appropriate b(t) and obtain the corresponding ν(t). A possible choice is

bφ(t) = 1 − k exp(−(t − T/2)2/σ2) , where 1/σ ≪ T is the characteristics time scale of

the operation; k is chosen to produce the desired phase shift. There is no fundamen-

tal limitation on the magnitude of σ, so the phase-shift operator can be implemented

indefinitely fast, even faster than 1/ν0 if the apparatus permits.

7.3.4 Nonlinear Operator

Nonlinear operators transform an annihilation operator to an operator involving quadratic

and higher order terms in a and a†. As discussed in Sec. 6.4.4, a nonlinear operator can

be achieved by applying a Hamiltonian that is at least third order of a and a†, such as

a Raman field with ω = 3ν0. The major issue of this approach is that the validity of

both the LDA and the RWA have to be satisfied, so the undesired component in the

Hamiltonian are suppressed. The consequence is that the power of the radiation field

is constrained, which limits the speed of the operation. Nevertheless, the current sep-

arate trap UBS architecture facilitates this laser-mediated nonlinear operator, because

the mode spectrum is simplified as only one phonon mode is exhibited in each trap.

Here I describe an alternative approach to realise a nonlinear operator by switching

on an additional quartic potential, i.e.

H4(t) = H0 + V4(t) ≡ H0 + F(t)x4 . (7.14)

In the interaction picture with respect to H0, the quartic potential becomes

V I4 =

F(t)~2

(2mν0)2

[

(6a†2a2 + 12a†a + 3) + a†2(4a†a+ 6)ei2ν0t + a†4ei4ν0t + h.c.]

. (7.15)

If the variation of F(t) is sufficiently slow, the off-resonant terms can be eliminated by

the RWA; the only effective terms are

HN ≡ F(t)~2

(2mν0)2(6a†2a2 + 12a†a+ 3) . (7.16)

Applying the quartic potential from t = 0 to T , the S-matrix of the operation in the

113


Schrodinger picture is given by

S4 = e−iµ(T )(6a†2a2+12a†a) , (7.17)

where

µ(t) =

∫ T

0

F(t′)~

(2mν0)2dt′ . (7.18)

I have neglected the unimportant global phase, and have employed the fact that [H0, HN ] =

0. Obviously, the addition of quartic potential realises a Kerr-like nonlinear phase shift.

The speed of this operation is mainly determined by the validity of the RWA. Accord-

ing to Ref. [67], applying the RWA is to collect the leading order terms in a series expan-

sion of time-averaged Hamiltonians. The sufficient condition for a valid series expansion

is that the largest eigenvalue of HN/~ should be much smaller than the off-resonant

frequencies, which are multiples of ν0 in our case. Although HN/~ has unbounded eigen-

values, the series expansion is still valid if the maximum phonon number in each mode,

nmax, is small. To estimate the RWA validity condition, I approximate F(t)~2/(2mω0)2

by ~µ(T )/T because V4 is slowly varying. The maximum eigenvalue of HN in the simu-

lation is hence n2maxµ(T )/T , which gives a valid RWA when

n2maxµ(T )

ν0T≪ 1 . (7.19)

7.4 Two-mode Operation

Because the ions are separately trapped in this UBS architecture, their modes are difficult

to be interact through laser operation. In this section, I describe a controlled collision

process that can implement a phonon beam splitter. The beam splitter would transform

the phonon modes of two separate ions according to Eq. (6.22). As discussed in Sec. 6.3,

universal bosonic simulation can be conducted by using the beam splitter and the single

mode operations discussed in Sec. 7.3.

The whole beam splitter process is summarized schematically in Fig. 7.3. Here I

consider the duration of the process is T , and it starts from t = −T/2. Harmonic wells

with moving trap centres are applied from t = −T/2 to transport the ions from the

storage traps to the pick-up distance and then switched off at t = −T ′/2, a double well

potential is immediately switched on to relay the transportation. The separation of the

double well shrinks and then expands. This action aims to bring the two ions to proximity,

so the two phonon modes can interact through the Coulomb interaction between the ions.

114


I

II

III

IV

V

Figure 7.3: Displacement of ions and variations of potentials during a phonon beamsplitter. The origin is defined as the mid-point between two storage traps. Step I, at−T/2 ≤ t ≤ −T ′/2, ions are transported by harmonic well from the storage traps to thepick-up positions. Step II, at t = −T ′/2, a double well is switched on to pick-up the ions.Step III, at −T ′/2 ≤ t ≤ T/2, the separation of double well shrinks and then expands.Step IV, at t = T ′/2 , the ions are brought back to the pick-up positions. Step V, atT ′/2 ≤ t ≤ T/2, the double well is switched off, and the harmonic wells pick-up the ionsand bring them to the storage traps.

The double well potential finally separates the ions to the pick-up distance. It is then

switched off at t = T ′/2, while two moving harmonic wells are switched on immediately

to transport the ions back to the storage traps at t = T/2.

In this section I set the origin of position, X = 0, to be the mid-point between the two

storage traps. I assume the system is both spatially and dynamically symmetric about

the origin. To simplify the discussion, the classical motion and the quantum fluctuations

are separated as 1

Xi ≡ Xi + qi ; Pi ≡ Pi + πi (7.20)

where the subscripts i = 1, 2 denote the ions involved in the beam splitter operation;

X and P are the position and momentum operator of the total motional state; Xi and

Pi ≡ m ˙Xi are the classical position and momentum of the ith ion; qi and πi are the

position and momentum operator of the quantum fluctuation.

The aim of the beam splitter is to transform the quantum fluctuations of the two

ions according to Eq. (6.22), whereas the ions will be classically stationary at the storage

1Formal procedure of separating the classical and quantum motion is referred to Sec. 4.3.1.

115


traps before and after the operation, i.e.

X1(−T

2) = X1(

T

2) = −L

2; X2(−

T

2) = X2(

T

2) =

L

2; (7.21)

P1(−T

2) = P1(

T

2) = P2(−

T

2) = P2(

T

2) = 0 . (7.22)

At the storage traps, the quadrature operators (position and momentum operators) of

the quantum fluctuation are the same as that of the phonon modes defined in Eq. (7.1),

i.e.

qi∣

∣

storage= xi ; πi

∣

∣

storage= pi . (7.23)

The core component of the beam splitter operation is a double well potential with

varying well separation. It can be constructed by a quartic potential, A(t)X4, and a

harmonic potential, B(t)X2, which can be realised in experiments [85]. The evolution

of the total motional state, |ψ〉, follows the Schrodinger equation i~∂t|ψ〉 = HBS(t)|ψ〉,where

HBS(t) =P 21

2m+P 22

2m+B(t)(X2

1 + X22 ) + A(t)(X4

1 + X42 ) +

e2

4πǫ0(X2 − X1). (7.24)

I note that the ions cannot tunnel pass each other due to the strong Coulomb repulsion,

so X2 > X1.

In terms of the variables in Eq. (7.20), the Schrodinger equation becomes

i~∂t|ψ〉 = (H0 + H1 + HB)|ψ〉 , (7.25)

where H0 and H1 collect the terms with the zero and the first order terms of the quadra-

ture operators, and HB contains the rest; |ψ〉 is the state of the quantum fluctuations.

The first termH0 = P 21 /2m+P 2

2 /2m+A(t)(X41+X

41 )+B(t)(X2

1+X21 )+e

2/4πǫ0(X2−X1)

is the total mechanical energy of the system; it contributes only an unimportant global

phase. The second term H1 vanishes if the classical equations of motion are satisfied, i.e.

˙Xi = Pi/m ; ˙Pi = −4A(t)X3i − 2B(t)Xi +

e2Xi

4πǫ0|Xi|(X2 − X1)2.

Because of the symmetry, we have X1(t) = −X2(t) and P1(t) = −P2(t). The classical

separation between the ions is defined as r ≡ X2 − X1, then the equation of motion is

116


reduced to the following:

r = −A(t)m

r3 − 2B(t)

mr +

e2

2πǫ0mr2. (7.26)

If the quantum fluctuation of position is much smaller than the separation of ions,

i.e.√

〈δq2〉/r ≪ 1, then HB can be approximated by a quadratic Hamiltonian, viz

HB ≈ H2 =π21

2m+π22

2m+

(

3

2A(t)r2 +B(t)

)

(q21 + q22) +e2(q2 − q1)

2

4πǫ0r3. (7.27)

The validity of this quadratic approximation will be discussed later.

Instead of analysing the motion of individual ions, it is simpler to study the collective

modes of motion, i.e., the centre-of-mass mode (+ mode) and the stretching mode (-

mode). The quadrature operators of the collective modes are defined as

q± =q2 ± q1√

2; π± =

π2 ± π1√2

. (7.28)

In this new basis, Eq. (7.27) can be written as the Hamiltonian of two decoupled harmonic

oscillators, i.e.,

H2 =π2+

2m+

1

2mν2+(t)q

2+ +

π2−

2m+

1

2mν2−(t)q

2− , (7.29)

where the mode frequencies are

ν+(t) =

√

3A(t)

mr2 +

2B(t)

m; (7.30)

ν−(t) =

√

ν2+(t) +e2

πǫ0mr3. (7.31)

The annihilation operators of the modes are defined as

A± =

√

mω±2~

q± + i

√

1

2~mω±π± . (7.32)

Because the collective modes are always decoupled, the variation of the double well

potential will induce only single mode squeezing or phase-shift on the collective modes.

Because we expect no parametric excitation after a beam splitter, the double well opera-

tion should give only a phase shift. In general, satisfying such condition would require us

to simultaneously solve the evolution of both modes. For simplicity, I assume the quartic

117


and the harmonic potentials are adjusted to produce a constant ν+, i.e.,

3A(t)

mr2(t) +

2B(t)

m= ν2+(t) = ν20 . (7.33)

According to Eq. (4.40), the + mode then remains unchanged after the operation, i.e.

A+ → A+.

The remaining problem is to construct a double well variation that only phase-shifts

the stretching mode, i.e. the annihilation operator transforms as A− → e−i2θA−, where

2θ = Θ(T ) − ν0T according to Eq. (4.40). The phonon modes of individual ions would

then transform as

A1 →1√2(A+ − e−i2θA−) = cos θA1 + i sin θA2 ; (7.34)

A2 →1√2(A+ + e−i2θA−) = i sin θA1 + cos θA2 , (7.35)

where an unimportant global phase e−iθ has been neglected. This transformation can

be rectified to the form in Eq. (6.22) by applying local phase-shift operators: before the

transformation, mode 2 is shifted with a phase −ieiφ′

; and after the transformation, mode

2 is shifted with a phase ie−iφ′.

The pick-up distance should be sufficiently large, at where ν− and ν+ are roughly the

same according to Eq. (7.31), i.e. ν−(T′/2) = ν−(T

′/2) = ν0. To avoid - mode from

parametric excitation, the ν−(t) should produce an auxiliary function b(t) that satisfies

b(−T ′/2) = b(T ′/2) = 1. Such a ν−(t) can be produced by controlling the quartic and

the harmonic potential strength, A(t) and B(t), while keeping ν+(t) as constant.

The appropriate time variations of A(t) and B(t) exist and are not unique; they can be

chosen in a manner that is convenient in practice. Here I outline a systematic procedure

to inversely engineer the desired A(t) and B(t) from an intellectually guessed b(t). Let

us consider b(t ≤ −T ′/2) and b(t ≥ T ′/2) are the necessary conditions for the operation

to be a phase-shift operator, but not a squeezing operator, on - mode. An ansatz that

these conditions are satisfied by construction is

bB(t) = 1− ke−t2/σ2

; (7.36)

where σ ≪ T ′ determines the time scale of the operation; the value k is chosen to generate

the desired phase shift.

With an ansatz of the auxiliary function, the corresponding ν−(t) can be deduced

inversely from Eq. (4.14). The time variation of the ion separation, r(t), is then obtained

118


from ν2−(t) by Eq. (7.31). A constraint on A(t) and B(t) is obtained by using the classical

equation of motion Eq. (7.26) and the time variation of r(t). Together with Eq. (7.33),

the unique form of A(t) and B(t) can then be deduced.

7.4.1 Ion Transport and Pick-up

Before and after the double well operation (step II-IV), the ions are transported back

and forth between the storage traps and the pick-up distance (step I and V). If both the

transporting harmonic potentials and the double well potential can be switched on and

off quickly, the pick-up process can be conducted smoothly that the phonon states will

not be disturbed. The pick-up distance is arbitrary; it can be chosen in a manner that

is convenient in experiments.

The ions’ classical velocity in the double well operation is determined by the choice of

A(t) and B(t). The velocity at the pick-up distance, ˙Xi(−T ′/2) and ˙Xi(T′/2), is obtained

by integrating the equation of motion Eq. (7.26) with the initial condition that velocity

vanishes at the turning point, i.e. r(0) = 0. In step I, the transporting harmonic wells

should increase the classical velocity of the ions from 0 at the storage trap to ˙Xi(−T ′/2)

at the pick-up distance. Similarly in step V, the transporting harmonic wells should

reduce the classical velocity from ˙Xi(T′/2) at the pick-up distance to 0 at the storage

traps.

During Step I and V, each ion is transported by a harmonic well with moving centre,

i.e., Eq. (4.6). In this case, the trap strength should be fixed, i.e., ν(t) = ν0, in order

to avoid parametric excitation. As discussed in Sec. 4.3.1 and 5.2, a moving harmonic

well will induce a classical displacement (position and momentum) on the ion. The

displacement depends on the path of the trap centre, si(t). By putting the variables in

Eq. (7.20) into Eq. (4.11), the classical equation of motion is

¨Xi(t) = −ν20(

Xi(t)− si(t))

, (7.37)

where the exact total classical displacement is given in Eq. (5.3). Appropriate si(t), which

produces the Xi and Pi that match the boundary conditions at t = −T/2,−T ′/2, T ′/2, T/2,

can be obtained by the inverse-engineering or bang-bang method [168], and further op-

timised for extra constraints [106, 46]. Such heatingless rapid ion transportation has

recently been realised by using segmented traps [175, 33].

During the transportation, the evolution of the quantum fluctuation is determined by

the Hamiltonian

HT =π2i

2m+

1

2mν20 q

2i . (7.38)

119


Obviously, the motional states will not be disturbed by HT and hence the transportation.

All in all, the operation from step I to step V realizes a phonon beam splitter, i.e.

Eq. (6.22), on the phonon modes in neighbouring storage traps.

7.4.2 Accuracy of beam splitter

Now let us consider the errors in the beam splitter operation. The transportation in step

I and V would cause small error if the harmonic well is sufficiently precise. The pick-up

process is also assumed to be fast enough that does not cause significant error. Such

a high precision and rapid switching of trap potential has recently been realised in ion

transportation experiments [175, 33].

Most of the error is expected to come from the double well process. One of the prob-

lem comes from the anharmonic terms in the Hamiltonian of quantum fluctuations. Let

us recall that the Hamiltonian H2 in Eq. (7.27) is constructed under quadratic approxi-

mation. In fact, the full Hamiltonian is given by

HB = H2−√2e2q3−

2πǫ0r3(r +√2q−)

+√2A(t)r(3q2+q−+ q3−)+

A(t)

2(q4++6q2+q

2−+ q4−) , (7.39)

which includes higher than second order (anharmonic) terms of quadrature operators.

When comparing with H2, the magnitude of these anharmonic terms is suppressed by

the ratio between the spread of quantum fluctuation and the ion separation. This ratio

can be characterised by the anharmonic factor:√

〈q2〉/r. According to the cases I have

studied, if a µs range beam splitter is conducted in a trap with ν0 in the MHz range, the

shortest ion separation is about l0 ≡ 3√

2e2/4πǫ0mν20 that is in the µm range 2. On the

other hand, the spread of quantum fluctuation is roughly,√n√

~/2mν0, which is about

tens of nm if the average number of phonon in a mode, n, is smaller than or at the order

of 10.

The influence of the anharmonic terms is numerically assessed by simulating a 50:50

beam splitter with the bB(t) in Eq. (7.36). The evolution of the motional states is

tracked by integrating the Schrodinger equation with the Hamiltonian in Eq. (7.39). For

numerical efficiency, the interaction terms between the + and - modes are replaced by

the expectation value, e.g. q2+ → 〈q2+〉. This is a good approximation in our case because

the back reaction scales as higher orders of the small anharmonic factor.

I consider the ions are 40Ca+ and the trap frequency of the storage traps is ν0 = 2π

MHz. The pick-up position is set as 50 l0/3√2 from the mid-point of the ions. The

2I note that when two ions are placed in a single harmonic well, the ion separation is l0.

120


Figure 7.4: Fidelity of the phonon state |n+〉|n−〉after a 50:50 beam splitter operationwith ν20σ

2 = 2, 3, 4, 5, 7, 9. Total time for the double well to bring ions from and back tothe pick-up position is 11, 13, 15, 17, 20, 22× 1/ν0 respectively. The minimum separationbetween ions is about 1 l0 in all the six runs. The dotted line shows a benchmark of 0.99fidelity.

operation speed is adjusted by tuning σ, and the value of k is chosen to generate the

desired phase shift.

I set the input states of both the CM mode and the stretching mode to be pure

Fock states, i.e. |ψ(−T/2)〉 = |n+〉|n−〉, such that the final states should be the same as

the input states up to a phase. Only the runs with n+ = n− are shown in Fig. 7.4 for

comparison, because the fidelity of the states with inhomogeneous phonon number, i.e.,

n+ 6= n−, is generally higher for the same maximum phonon number, i.e., maxn+, n−.The double well process generates less than 1% error when ν20σ

2 & 5 for nmax ≤ 8; the

total process time is only about 17/ν0 ≈ 2.7 µs. In addition to the fidelity, I have also

examined the accuracy of the final state. In all cases simulated, the phase errors are less

than 1%. These simulated results suggest that in a bosonic simulation with single digit

number of phonons in each mode, a quality phonon beam splitter can be implemented

within a few µs.

The accuracy of the beam splitter is worsened, as expected, when more phonons are

involved, because the anharmonic factor increases. Besides, a higher operation speed also

exacerbates the error due to two reasons. Firstly, a faster beam splitter would require

a stronger Coulomb force to interact the phonon states, so the ions have to be brought

closer and hence the anharmonic factor would be larger. Secondly, when considering

121


Eq. (7.39) in the interaction picture with respect to H2, the terms that are third order

to√

〈q2〉/r are off-resonant. The contribution of these terms is suppressed by the RWA

if they are slowly varying, then the effective Hamiltonian would be reduced to the fourth

order of the anharmonic factor. On the other hand, the off-resonant terms are significant

for a high speed operation because the RWA is less effective.

I note that the actual anharmonic effect crucially depends on the time variation, as

well as the physical implementation of the trap potential, e.g., the configuration of the

electrodes. In practice, the anharmonicity can be suppressed by optimising the potential

with a better design of trap and of the ansatz of b(t). In any case, the above numerical

analysis provides evidence that the beam splitter operation is accurate even the potential

is realistically anharmonic.

7.5 Initialization and readout

The single 3 mode state initialisation techniques presented in Sec. 6.6 is still applicable

in the current separated trap architecture. In addition, Gaussian states can also be effi-

ciently created by varying the trap potential, i.e., by implementing Gaussian operations

on the motional ground state. However, the creation of non-Gaussian states by using the

the nonlinear phase gate in Sec. 7.3.4 may require a complicated sequence of operations.

Some non-Gaussian states, such as Fock states and Schrodinger’s cat states, can be more

efficiently created by laser interaction.

When comparing with the single trap architecture in Ch. 6 , the current separated trap

architecture simplifies the laser-mediated state initialization. Since each trap contains

only one mode, other modes would not be accidentally excited, so some conditions of

RWA can be relaxed. Besides, it is no longer necessary to implement techniques, such

as composite pulses and shielding, to prevent the laser operations from influencing other

ions.

Information about the phonon states can be extracted by the three measurement

schemes suggested in Sec. 6.5: adiabatic transfer, post-selection techniques, and using

multiple electronic states. In all of these schemes, the current separated trap architecture

is more favourable than the architecture in Ch. 6. Because the ions are individually

trapped, the recoil of an ion after fluorescence measurement does not distort other phonon

modes. The spectral distribution of resonance is also simplified because each trap contains

only one mode; the speed of sideband transition is hence increased as a stronger pulse

3in Sec. 6.6, the creation of multiple mode states involves laser-mediated beam splitter operation,which is not applicable in the current architecture.

122


can be applied without accidentally mixing other modes.

7.6 Conclusion

In this chapter, I have described a possible architecture to implement the universal

bosonic simulator by using separately trapped ions. The excitation of an ion’s quantized

motion can simulate a bosonic mode. Linear single mode operations can be realized by

changing the strength and the centre of the trap harmonic potential. Nonlinear phase-

shift operator can be implemented by exerting a perturbative quartic potential. Linear

beam splitter is implemented by controlling the ion separation through varying a double

well potential; the interaction between phonons ensues from the Coulomb interaction

between the ions. By alternatively applying these operators, arbitrary bosonic evolution

can be efficiently simulated [116, 117].

Provided that the harmonic potential implemented in experiment is sufficiently ac-

curate and controllable, there is no fundamental limit on the speed of the single mode

linear operators, and a linear beam splitter can be implemented within tens of 1/ν0 if

the phonon number in each interacting mode is about or less than 10.

I end this chapter with some discussions on the capabilities of the ion trap UBS.

When comparing with conducting linear optical experiments, the ion trap UBS has both

strength and weakness. Firstly, passive linear elements, which include beam splitters and

phase-shift operator (wave plate), as well as displacement operator, have been accurately

implemented in optical system. The phonon counterparts that require laser operations or

manipulations of trap potential may include more error than in the optical system. On

the other hand, applying a squeezing operation on an arbitrary optical state is difficult

to implement, but squeezing an ion motional state can be achieved with the same level

of accuracy as other linear elements. Furthermore, optical nonlinear effect is small and

highly dependent on the material of the nonlinear optical element, while the nonlinear

potential applied on phonon is tuneable and there is no fundamental limit of its strength.

Therefore, in terms of the operation, ion trap UBS is not overwhelming advantageous

over optical systems.

The advantage of an ion trap UBS can be seen in the state initialisation and readout.

Most of the quantum states prepared in optical experiments are Gaussian states, while

the preparation of non-Gaussian state is inefficient, due to the large amount of post-

selection required or the general weakness of optical nonlinearity. On the other hand,

non-Gaussian states, such as Fock states and Schrodinger’s Cat state, have been realised

in ion trap experiments with high fidelity [127, 134]. In general, arbitrary ion motional

123


states can be deterministically created by applying a sequence of resonant pulses [69].

Such techniques would be difficult to realise in optical systems, even in foreseeable future.

In most optical experiments, quantum states are measured by photon non-resolving

detectors, most of which the efficiency is about 10 − 90% [76]. On the other hand,

both the electronic-motional state transition and electronic state measurement have been

implemented with over 90% accuracy in ion trap experiments [139] 4. Furthermore, the

readout scheme described in Sec. 6.5 could construct measurement schemes, such as

quantum non-demolition measurement and arbitrary binary PVM, that are difficult to

deterministically implement in optical systems. With the possibility of creating non-

Gaussian states and new measurement schemes, ion trap UBS provides a new testbed

for studying quantum optical problems and quantum information protocols.

The two architectures introduced in Ch. 6 and the current chapter have their own

strength and disadvantages; their practicality depends on the simulation to be conducted.

All the necessary techniques for implementing the “multiple ions in single trap” UBS

have been demonstrated in experiments. Besides, the accuracy of each operation is high

because the intensity and frequency of laser field can be precisely manipulated, while the

trap potential is kept constant throughout the simulation. However, the speed of each

operation is limited by the validity of LDA and RWA, in order to engineer the desired

form of interaction and to avoid unwanted influence on other modes. The condition of

RWA is more stringent when more ions and modes are involved in the simulation.

On the other hand, the “single ion in individual trap” approach is more scalable,

because the quality of each operation is independent of the number of ions in the UBS.

More explicitly, if a single mode bosonic operation is applied by exerting focused laser

field, there is low probability that other modes are influence because the ions are sep-

arately trapped. Apart from the laser-mediated operation, each bosonic operation can

also be realised by rapidly tuning the trap potential, which the speed is not limited by

the validity of LDA and RWA. In addition, the measurement of each mode would not

affect the state of other modes. This property facilitates the simulations that involves

post-selection or monitoring during the process. The essential techniques for implement-

ing this architecture is the high precision and speed manipulation of the trap potential,

which has been realised in recent experiments. The quality of the bosonic operations

implemented by these techniques has to be further investigated.

In both UBS architectures, nonlinear operations can be simulated efficiently but rela-

4Because the motional state cannot be directly measured, the accuracy of sideband transition canonly be observed indirectly. For a blue sideband transition, the accuracy can be deduced from the |g〉population after the pulse. As shown in Ref. [93], the experimental result is less than 10% deviated fromideal.

124


tively slow. A simulation can be much faster if it involves only linear operations. Various

interesting bosonic phenomena can be investigated by using only linear operations, in

particular the boson-sampling idea proposed by Aaronson and Arkhipov [17]. A boson-

sampler is an array of linear bosonic elements with Fock state inputs and boson number

non-resolving detection at the output. The authors claim that if there exists a classical

algorithm that efficiently samples the probability distribution of the output detection,

then the polynomial hierarchy would collapse to the third level, which is generally be-

lieved to be impossible in computer science [11]. In other words, if a boson-sampler were

realised, it would be a machine that exhibits post-classical computing power. Aaronson

and Arkhipov suggest that a meaningful demonstration of boson-sampler would require

n = 10 to 50 bosons and about n2 to n5 log n modes. Due to the high scalability, fast

and high quality operation, ability of deterministic Fock state preparation, and the flex-

ible and accurate measurement, ion trap bosonic simulator is a promising platform to

implement boson-sampling, and thus to demonstrate the post-classical computing power.

125

Chapter 8

Rapid ion re-cooling by swapping

beam splitter

8.1 Introduction

The ion trap system is regarded as the most advanced implementation of quantum com-

puters (QC) [50, 32], where various building blocks [53, 143] , such as fast and precise

quantum gates [35, 111], long time storage of quantum information [103], and high fi-

delity readout [139], have been demonstrated experimentally. In general, achieving high

fidelity logical operations requires minimal motional excitation of the ions. However dur-

ing computation, the ions are unavoidably heated up by, for example, fluctuations of

the trap potential, imprecise ion transportation, and momentum gain from recoil during

fluorescence readout. In practice, the ions have to be frequently re-cooled by sympathetic

cooling [98], which takes about hundreds of microseconds (µs) [93]. The cooling time has

recently been improved to tens of µs by using electromagnetically-induced-transparency

techniques [138, 137, 115]. However, the cooling process remains a speed bottleneck of an

ion trap QC because its duration is an order of magnitude longer than other operations

[78]. This speed bottleneck should be resolved for a faster ion trap quantum computer

for better preservation of quantum coherence and higher computational power. A recent

proposal suggests that µs range cooling can be achieved by using sequences of strong

laser pulses [121]. However this method is subjected to the limitation from the laser’s

power.

In this chapter, I describe a scheme that can rapidly re-cool a pair of ion qubit

without applying laser cooling during the computation. The scheme is divided into three

processes as shown in Fig. 8.1. Firstly, coolant ions, i.e. ions that will not be involved

126

Chapter 8. Rapid ion re-cooling by swapping beam splitter

in quantum logic gates and can be different in species from the qubits, are prepared in

the motional ground state before quantum computation. Each coolant is stored in an

individual harmonic well inside a linear trap.

The second process, which is the core of the cooling scheme, is a swapping beam splitter

(SBS). As an extension of the phonon beam splitter described in Sec. 7.4, here the SBS

can swap the motional states of two ions even they have different masses. When a qubit

has to be re-cooled, it is brought to the linear trap of a coolant. A SBS is then applied

to transfer the motional ground state from the coolant to the qubit, thus the qubit is

effectively cooled. If excessive coolants are prepared, a new coolant can be employed in

each round of cooling, thus laser cooling is not required during the computation.

The last process is to combine the individually cooled qubits to a ground state qubit

pair, which can be done by a diabatic ion combination process. Both the SBS and the

ion combination can be implemented by a controlled collision of the ions in a precisely

manipulated double well trapping potential. I will show that these processes can take

less than ten trap oscillation periods, so the total process durations are at the µs range

for state-of-the-art MHz traps. The necessary rapid and precise control of a double well

potential has been demonstrated by using micro-fabricated surface traps [33].

8.2 Model

As discussed in Sec. 4.3.1, if an ion is transported through a linear trap by a moving

harmonic well with fixed strength, the ion’s quantum fluctuation is unchanged while

only a classical displacement is induced. In other words, linear ion transportation can be

diabatic, i.e., arbitrarily fast without causing any motional excitation, if the harmonic

well centre is controlled so that the final displacement vanishes [168, 106, 46, 175, 33]. So

let us consider the ground state coolants are restricted to move only in their respective

linear traps. On the other hand, the heated qubit has to be transported, possibly through

linear traps and junctions, to the coolant’s trap for cooling. The transportation would

further heat up the qubits, but they will eventually be cooled.

A SBS is implemented by a controlled collision of the two ions (see Fig. 8.1). The

ions are radially tightly confined but weakly trapped in the axial (x) direction. The total

axial motional state of the ions |Ψ〉 is governed by the equation i~∂t|Ψ〉 = H|Ψ〉, where

H =

2∑

i=1

( P 2i

2mi+

1

2ξ2i (t)

(

Xi − Ri(t))2)

+e2

4πǫ0(X1 − X2).

127


Heated ion

Cooled ion

Temperature in transient

(I)

(II)

(III)

Figure 8.1: Outline of the cooling process. Step I: Each heated (red) qubit (large ball)is transported to one of the coolants’ traps to collide with a ground state (blue) coolant(small ball). In a linear trap, the ions are transported by moving harmonic well untilpicking up by the double well potential. Step II: The double well potential varies theion separation to implement the SBS. The qubit’s motional excitation is transferred tothe coolant. Step III: Moving harmonic wells pick-up the ions from the double well.The heated coolant is to be discarded or re-cooled. Two individually cooled qubits canbe combined into a single harmonic well for entanglement operation. The combinationprocess can be fast and can cause negligible excitation.

128


Here X and P denotes the position and momentum operator of the total motional state.

The qubit (ion 1) and the coolant (ion 2) can be different in mass, i.e., m1 6= m2.

Without making any assumption on the trap implementation, I characterize the local

trap potential by a displaced harmonic well on each ion. The four local trap parame-

ters, ξ21(t), ξ22(t), R1(t), and R2(t), can be determined when expanding the global trap

potential around the classical position of each ion. The realistic implementation of such

potential will be discussed in the Sec. 8.7. Time variation of these parameters, which

can be independently controlled by tuning the global potential, will be specified by four

constraints that leads to a SBS operation.

The motional excitation is characterised by the quantum fluctuation around the clas-

sical displacement of the ions. By definition, a cooling is complete only if the quantum

fluctuation is brought to the ground state while the ion has no classical displacement.

Let us define the state of the quantum fluctuation as |ψ〉 ≡ D†1(x1, p1)D

†2(x2, p2)|Ψ〉,

where Di(xi, pi) = exp(

i(xiPi − piXi)/~)

is the same displacement operator defined in

Eq. (4.8); xi and pi are classical parameters that could be chosen as the classical position

and momentum of ion i. After neglecting a constant term that contributes only a global

phase, the state |ψ〉 obeys the equation

i~∂t|ψ〉 = (H1 + H2)|ψ〉 . (8.1)

H1 involves only the first order position and momentum operators, i.e., H1 = V1p1 +

V2p2 + F1q1 + F2q2, where

Vi =pimi

− xi ; Fi = pi + ξ2i (t) (xi − Ri(t)) +(−1)ie2

4πǫ0r2. (8.2)

r ≡ x1−x2 > 0 is the ion separation. The above equations become the classical equation

of motion when Vi = 0 and Fi = 0, thus H1 = 0 for my choice of xi and pi.

The dynamics of the quantum fluctuation is governed only by H2 that involves second

and higher order of operators:

H2 =2∑

i=1

( p2i2mi

+1

2ξ2i (t)q

2i

)

+e2

4πǫ0r3(q1 − q2)

2 +O(q3) . (8.3)

For clarity, I have recast the position and momentum operators of the quantum fluc-

tuation as q and p respectively, and the “motional state” is only referred to that of

quantum fluctuation hereafter. The Coulomb potential is Taylor-expanded with respect

to (q1 − q2)/r. For the moment, quadratic approximation is applied on H2, i.e., O(q3)

129


is neglected. This approximation is applicable in our case because the spread of ions’

wavefunction is much shorter than the ion separation, i.e.,√

〈q2i 〉 ≪ r. The validity of

this approximation will be further examined in Sec. 8.8.

8.3 Cooling

It is well known that energy can be transferred between harmonic oscillators in the

presence of weak coupling [12]. Such effect is recently demonstrated with two separately

trapped ions, of which the motional state is swapped by the Coulomb interaction [36, 79].

The cooling scheme discussed in this chapter employs a similar effect to transfer the

motional excitation from the qubit to the coolant. The major difference here is that both

the interaction and local trap strength are tuneable, and the evolution of the harmonic

oscillators is solved exactly instead of perturbatively. As we will see, these tricks allow

the current proposal to have the following improvement: (i) energy can be completely

transfer between two oscillators even they have different masses; (ii) energy transfer can

be diabatic by using strong coupling.

Within the quadratic approximation, the evolution under H2 is a two-mode squeezing

operation on the ions’ motional state [117, 176]. The squeezing parameters depend on

the tuneable local trap strength, ξ2i (t), and on the Coulomb coupling that is determined

by r, which is controllable by adjusting Ri(t). I will show a systematic way to obtain the

trap parameters that the two-mode squeezing becomes a SBS, i.e., after the process at

0 < t < T the annihilation operators transform as

U †T a1UT = a2e

iθ ; U †T a2UT = a1e

iθ , (8.4)

where

ai ≡(√√

miξi(0)

2~qi + i

√

1

2~√miξi(0)

pi

)

; (8.5)

Ut is the evolution operator at time t. The phase factor θ does not involve in the cooling

process. The local trap strength before and after the SBS are the same, i.e., ξ2i (0) = ξ2i (T ),

so the initial and final states can be characterised by the same ai.

To see how a SBS cools the qubit, let us consider the initial motional state of the

coolant is the ground state while that of the qubit is an arbitrary pure state, i.e., |ψ(0)〉 =∫

f(α)|α〉1dα ⊗ |0〉2, for some complex function f . This state lies in the eigensubspace

of the coolant’s phonon number operator: a†2a2|ψ(0)〉 = 0. According to Eq. (8.4), the

SBS transforms the eigenvalue equation as a†1a1|ψ(T )〉 = UT U†T a

†1UT U

†T a1UT |ψ(0)〉 =

130


UT a†2a2|ψ(0)〉 = 0. This derivation implies that the qubit will result in the ground

motional state. Since the eigenvalue equation is valid for any complex function f , the

SBS can cool a qubit with any initial motional state.

8.4 Swapping Beam Splitter

The construction of a SBS is clearer when considering the collective modes of the quantum

fluctuation. Let us define the quadrature operators of the centre-of-mass (+) mode and

the stretching (-) mode as

q± ≡ 1√2(q1 ±

√

m2

m1q2) ; p± =

1√2(p1 ±

√

m1

m2p2) . (8.6)

Then in the quadratic approximation as H2 can be re-written as

H2 ≈p2+2m1

+p2−2m1

+1

2m1ν

2+(t)q

2+ +

1

2m1ν

2−(t)q

2− + E q+q− , (8.7)

where the coupling strength between the modes is

E =ξ21(t)

2− m1

m2

ξ22(t)

2+(

1− m1

m2

) e2

4πǫ0r3, (8.8)

and the mode frequencies are

ν2±(t) =ξ21(t)

2m1

+ξ22(t)

2m2

+(

√

1

m2

∓√

1

m1

)2 2e2

4πǫ0r3. (8.9)

Since the Hamiltonian in Eq. (8.7) involves only the quadratic terms of the quadrature

operators, the evolution is generally a two-mode squeezing operation. Here I discuss the

necessary conditions that correspond to a SBS. Let us define the annihilation operators

of the modes as

a± ≡√

m1ν02~

q± + i

√

1

2~m1ν0p± , (8.10)

where ν0 ≡ ξ1(0)/√m1 is the qubit’s initial trap frequency. The annihilation operators

of the collective modes and the ions’ motional state are related as a± = (a1 ± a2)/√2.

Since a beam splitter should not induce parametric excitation, the modes should only

be phase-shifted after the process, i.e., U †T a±UT = a±e−iθ±(T ). Additionally, the beam

splitter is a SBS, i.e. Eq. (8.4) is satisfied, if the modes acquire a π phase difference after

the operation, i.e., θ−(T )− θ+(T ) = π.

131


The system involves two coupled time dependent harmonic oscillators, of which the

analytical general solution is difficult to obtain. Here I provide a procedure to systemati-

cally deduce the time variation of the four local trap parameters that would yield a SBS.

Firstly, I require the + and - modes are decoupled, i.e. E = 0. According to Eq. (8.8),

this can be achieved by imposing Constraint 1 as:

ξ22(t) =m2

m1ξ21(t) +

(m2

m1− 1) e2

2πǫ0r3. (8.11)

The system is reduced to two decoupled harmonic oscillators, where the mode fre-

quencies can be individually controlled by tuning ξ21(t) as well as Ri(t) that control r.

While appropriate mode frequencies can be individually obtained by using the inverse-

engineering method (to be discussed), one particular solution is to set one of the mode,

e.g. + mode, to be time independent, i.e., ν+(t) = ν0. According to Eqs. (8.9) and (8.11),

this imposes the Constraint 2 as:

ξ21(t) = m1ν20 −

(

1−√

m1

m2

) e2

2πǫ0r3. (8.12)

Then + mode is not parametric excited and the phase is θ+(t) = ν0t.

With Constraint 1 and Constraint 2 satisfied, the ion separation is then uniquely

determined by ν2−(t) via the Constraint 3:

ν2−(t) = ν20 +e2√

m1m2πǫ0r3. (8.13)

The remaining problem is to find an appropriate ν2−(t) so that - mode is not parametric

excited and acquires the desired phase. Here I introduce an inverse engineering method

to find such an ν2−(t). As a time dependent harmonic oscillators, the evolution of -

mode can be exactly solved by using the dynamic invariant formalism as discussed in

Sec. 4.3.1. At any time t, the annihilation operator of - mode transforms as Eq. (4.30).

These parameters are uniquely determined by a real scalar auxiliary function b(t) that

satisfies Eq. (4.14), i.e.,

b(t) + ν2−(t)b(t)−ν20b3(t)

= 0 , (8.14)

with b(t < 0) = 1. Parametric excitation is absent if η∗(t > T ) → 1 and ζ(t > T ) →0. This imposes a boundary condition on the auxiliary function: b(t > T ) → 1. An

additional boundary condition is required on b(t) that yields the desired final phase

θ−(T ) = θ+(T ) + π = ν0T + π.

132


In order to obtain the local trap parameters that realise a SBS, let us adopt an ansatz

for b(t) to obtain an appropriate ν2−(t). The ion separation, r(t), is then determined by

Constraint 3. ξ21(t) and ξ22(t) are obtained by Constraint 2 and 1 respectively. For

R1(t) and R2(t), one constraint is already imposed by the desired r(t) and the classical

equation of motion, Eq. (8.2); an additional constraint on the classical centre-of-mass

motion is needed to fix the parameters. A possible choice of this Constraint 4 is the

symmetric ion motion: x1(t) = −x2(t) = r(t)/2.

8.4.1 Ansatz

I here suggest a class of ansatz of b(t) that all the boundary conditions are satisfied at

construction:

b(t) =

(√π

ν0σe−(t−0.5T )2/σ2

+ 1

)−1/2

. (8.15)

The speed of the SBS is determined by the parameter σ. For ν20σ2 = 2 and 3, the SBS

process time, T , is 8.3 and 10.2 trap oscillation periods respectively 1. I note that the

scaled process time, ν0T , is independent of the ions’ mass and ν0, and does not affect

the quality of cooling within the quadratic approximation of H2. Therefore, the qubit

cooling time is generally in the µs range if the trap frequency is a few MHz.

As an example, a controlled collision between a 40Ca+ qubit and a 24Mg+ coolant

with ν0 = 2π MHz was simulated. The cooling time is T = 1.3µs for ν20σ2 = 2. Time

variation of mean phonon number, ion separation, and local trap parameters are shown

in Fig. 8.2.

8.5 Ground state qubit pair

Now I discuss how to use the above cooling scheme to prepare a ground state qubit

pair for high fidelity quantum logic operation. Here I specifically consider the KMW

quantum computer architecture that is constituted by numerous interconnected traps

(c.f. Sec. 4.3), though the method is also applicable to other architectures that ions are

movable in linear traps, e.g. Ref. [135]. Let us consider two qubits are transported to the

linear trap that contains an array of individually trapped coolants (Fig. 8.3a). The qubits

can be individually cooled by three rounds of SBS (Fig. 8.3b I-III). Between each round,

the qubits (coolants) are transported by moving harmonic wells with strength m1ν20

1For numerical reason, T is defined at when the total phase difference is at least 10−4 deviated fromπ, i.e., |θ

−(T ) − θ+(T ) − π| ≤ 10−4. With this setting, the qubit possesses no more than 10−6 final

motional excitation at t = T .

133


- 4 - 2 0 2 40

2

4

6

- 4 - 2 0 2 40

2

4

6

8

10

- 4 - 2 0 2 4 - 10

- 5

0

5

10

- 4 - 2 0 2 40.

0.5

1.

1.5

(a) (b)

(c) (d)

Figure 8.2: (a) Let the qubit be initially in a thermal state with 〈a†1a1〉 = 5. The meanphonon number of qubit (solid) and coolant (dashed) is swapped after the SBS. Timevariation of (b) ion separation, (c) R1 (solid) and R2 (dashed) , (d) ξ

21 (solid line) and ξ22

(dashed line), for the SBS following the ansatz in Eq. (8.15) with ν20σ2 = 2. All the length

in the figure is expressed in the unit of the characteristic length, l0 =3√

2e2/4πǫ0m1ν20 ≈

5.61µm, which is the separation of two 40Ca+ ions in a single harmonic well with ν0 = 2πMHz.

134


(a)

Q1 Q2 C1 C2(b)

II

III

IV

I

Figure 8.3: (a) Heated qubits are transported to the linear trap containing individuallytrapped coolants. (b) Sequence of constructing ground state qubit pair: (I) Motionalexcitation is transferred from qubit Q2 to coolant C1 through a SBS. (II) Two SBS aresimultaneously conducted to swap the motion between qubit Q1 and Q2, and betweencoolant C1 and C2. (III) Repeat the procedure in I. Both Q1 and Q2 are cooled after thisstep. (IV) The qubits are combined in a single harmonic well through the heatingless ioncombination process.

(m2ν20). As discussed in Sec. 4.3, the ions’ classical motion can be freely manipulated

by precisely controlling the harmonic well, while the quantum fluctuation is unaffected

[168, 106, 46, 175, 33].

After two qubits are individually cooled, they have to be combined in a single har-

monic well (see Fig. 8.3b IV) for entanglement operation. The ion combination can be

rapid and causing minimal excitation. This heatingless ion combination can be viewed

as half of a SBS, which the double well potential stops varying when it converges to a

single harmonic well. Time variations of trap parameters in this process can be obtained

by a similar procedure as in SBS. Here, Constraint 1 is not necessary as the + and

- modes are decoupled when both qubits have the same mass. This constraint can be

modified to require symmetric local trap strength, i.e., ξ21(t) = ξ22(t). Constraint 2,

which requires constant + mode frequency, can be retained so that the + mode is not

parametric excited during the combination. Constraint 3 is still required to yield the

desired - mode frequency by inverse engineering. Constraint 4 can still be imposed to

require symmetric motion.

For the ansatz of b(t), b(t < 0) → 1 is remained because the initially separated ions are

not excited. The crucial difference here is the boundary condition of b(t > T ). According

to Eq. (8.14) and the fact that ν2− = 3ν20 when two ions are trapped in a single harmonic

well [90, 107], the steady value of the auxiliary function after the combination should be

b(t > T ) → 3−1/4. The - mode will not be parametric excited if b(t) remains constant for

135


I

II

III

Figure 8.4: Variations of potentials during diabatic ion separation. Step I, a quarticpotential is added to the common trap to form a double well potential. Step II, thedouble well potential expands to separate the ions, until they reach the pick-up positions.Step III, harmonic wells pick-up the ions and bring them to other traps.

t > T , which is a necessary condition that the ion combination process does not cause

any motional excitation [113]. An example of ansatz of such b(t) is

b(t) =

(

14√3− 1

)

e(t−0.5T )3/σ3

+ 1 , (8.16)

As an illustration, if the initial separation between two 40Ca+ ions is r(0) = 100l0, they

can be combined to a single well, i.e., r(T ) = l0, in T ≈ 5/ν0 ≈ 0.8µs for ν0 = 2π MHz

and an ansatz with ν0σ = 2.

I note that the above ion combination process can be reversed to diabatically separate

an ion pair. While all the constraints are the same (with the same flexibility that they

can be chosen according to practical needs), the only difference is that the boundary

conditions of b(t) are reversed, i.e., b(t ≤ 0) = 3−1/4 and b(t ≥ T ) = 1. Any ansatz

that implements a diabatic combination can be modified for the separation process by

setting t − T → T − t. The diabatic ion separation process is applicable after the ions

have conducted entanglement operation in a single trap. Since the motional excitation

is reduced, subsequent cooling time is shortened and the speed of quantum computation

is hence improved. The whole process is summarized in Fig. 8.4.

8.6 Transport between traps

As shown as Step I (III) in Fig. 8.1, the ions are picked-up by (from) the double well

before (after) the SBS. The classical motion of the ions during the SBS is specified by the

ansatz of b(t). Before the SBS, the ions have to be transported to the specified pick-up

positions, xi(0), and accelerated to the specified pick-up velocities, xi(0). Similarly after

the SBS, the ions have to be transported away from xi(T ) and decelerated from xi(T ).

As discussed in Sec. 7.4.1, the ions can be transported by moving harmonic potentials

136


with fixed strength to avoid parametric excitation [168, 106, 46]. For a smooth transition

between the double well and the moving harmonic potential, the local trap parameters

should be continuous at t = 0 and T . This requires the trap strength of the harmonic

wells to be ξ21(t) = ξ20 and ξ22(t) =m2

m1ξ20.

If the trap potential can be changed sufficiently rapidly and accurately, the speed of

ion transportation is not limited at the range of the harmonic oscillation period [106].

As demonstrated in recent experiments, ions in a MHz trap can be transported in a few

µs without significant heating [175, 33]. The error of the transportation stage is thus not

likely to reduce the performance of the cooling process at the current level of technology.

8.7 Implementation of potential

In the above discussion, the trap potential is represented by two displaced harmonic wells,

i.e., Eq. (8.1). This representation is an approximation of the global trap potential around

the classical position of the ions. In practice, such local potential can be implemented

by various kinds of global potential, the choice of which depends on the experimental

convenience.

Here I give an example of a feasible implementation

V (x) = E0(t)x+ α(t)x2 + β1(t)(x− x0)4 + β2(t)(x+ x0)

4 . (8.17)

The first two terms denote a harmonic potential with variable trap strength and centre,

which can be realised by segmented traps (e.g. [33]); the last two terms are two quartic

wells with fixed centre at x0 and −x0 respectively, which can be realised by two sets

of octupole potentials [85]. By Taylor-expanding V (x) around the vicinity of the ions’

classical position, x1 and x2, and collecting the first and second order terms, at any time

t the local and global trap parameters are related as

1 2x1 4x31− 4x31+

1 2x2 4x32− 4x32+

0 2 12x21− 12x21+0 2 12x2x− 12x22+

·

E0

α

β1

β2

=

(x1 −R1)ξ21

(x2 −R2)ξ22

ξ21ξ22

,

where xi± ≡ xi ± x0. If the desired time variation of the local trap parameters, ξ21(t),

ξ22(t), R1(t), and R2(t), are known, then the time variation of the global trap parameters,

E0(t), α(t), β1(t), and β2(t) can be obtained by inverting the above equation.

137


8.8 Anharmonicity

In the above discussion, I have shown that a SBS can perfectly cool a qubit in case of the

quadratic approximation, i.e., the model of coupled harmonic oscillators in Eq. (8.3) is a

good approximation to the full Hamiltonian. In practice, the local potential experienced

by the ions is not purely harmonic; the anharmonicity would induce motional excitation

during the SBS.

The anharmonicity comes from both the global trap potential and the Coulomb in-

teraction. The global trap anharmonicity highly depends on the configuration of the

experiment. For example, if the ion motion is symmetric and the applied global poten-

tial is a fourth order polynomial of position, i.e. V (x) =∑4

j=1 Vjxj , where Vj are real

parameters related to ξ’s and R’s, then the leading order anharmonic terms in Eq. (8.3)

are at the third order of q’s, viz. [(ξ21 − ξ22)/6r + (ξ21R1 − ξ22R2)/r2]q31 + [(ξ21 − ξ22)/6r −

(ξ21R1 − ξ22R2)/r2]q32. According to my simulation, the motional excitation caused by

these terms can be a few times higher than that caused by the Coulomb anharmonicity.

However, these third order terms can be suppressed if the applied potential is a higher

order polynomial of x. In practice, the global trap anharmonicity can be suppressed by

optimising the geometry and the potentials of the electrodes [85, 19].

On the other hand, the anharmonicity of the Coulomb potential are the higher order

terms in the Taylor expansion in Eq. (8.3). Such terms represent the nonlocal interaction

between ions that could not be fully suppressed by adjusting the trap potential. Each

order of the anharmonic terms roughly scales as√

〈q2〉/r, so the anharmonicity becomes

significant if the mean initial excitation 〈n1〉in is higher or the ion separation is shorter.

As a brief estimation, the minimum ion separation is about l0, so the scale factor is

roughly√

〈n1〉in√

~

2mν0/l0, which is at the range of 0.01 if the mean phonon number is

about or less than 10.

The effect of Coulomb anharmonicity on the cooling performance was assessed in

more details. Eq. (8.1) is numerically integrated when the lowest order term in O(q3) in

Eq. (8.3), e2

4πǫ0r4(q1 − q2)

3, is included. Although the anharmonic heating is expected to

be more serious in a faster SBS, because the ions should be brought closer for a stronger

Coulomb interaction, my numerical results show that the heating effect is more sensitive

to the ansatz of b(t) rather than only the speed.

Fig. 8.5 shows the simulation result for three cooling process with different ansatz.

Case I and Case II correspond to the ansatz in Eq. (8.15) with ν20σ2 = 2 and 3. The ansatz

of Case III is chosen as b−(t) = 1/√

γ exp(−t4/k′) + 1 with k′ = 8/ν40 and γ ≈ 3.048. The

process time of the three cases are respectively T ≈ 8.3/ν0, 10.2/ν0, 6.3/ν0. The numerical

138


Figure 8.5: Final motional excitation of the qubit, 〈n1〉f , induced by the Coulomb an-harmonicity when the two ions are initially prepared in a Fock state, |nin〉1|0〉2, and thenundergo the SBS process specified by Case I (blue), II (red), III (brown).

results show that higher excitation is induced in Case I than in Case III although the

former one is slower. This result could be understood from the fact that the minimum

ion separation in case I is shorter than that of case III. In either case, our result shows

that even if the qubit initially has 40 phonons, the Coulomb anharmonicity induces no

more than 10−3 phonon on the qubit. Therefore the anharmonicity of potential is not

deemed a serious threat to the performance of the SBS cooling process.

Similar to the phonon beam splitter, the error of the diabatic ion combination method

is produced by the anharmonicity of the global trap potential and the Coulomb interac-

tion. To demonstrate the feasibility of the scheme, the evolution equation Eq. (8.1) is

again numerically integrated when the lowest order term in O(q3) included. The ansatz

of b(t) is taken as Eq. (8.16). I consider the ions are Ca+ and ν0 = 2π MHz, where the

final separation r(0) = l0. After combining the ions from about 80 l0 to l0, which the

duration is about 5×1/ν0 ≈ 0.8µs, less than 0.001 quanta is excited for both the + mode

and the - mode.

8.9 Fluctuation

Practical implementation of the trap potential may involve imperfection. Here I numer-

ically assess how the cooling performance would be affected by the inaccuracy of the

139


global trap potential. I employ the implementation of potential in Sec. 8.7. For each of

the global trap parameters, a random Gaussian error is added. The characteristic width

and time of the Gaussian error are denoted by κ and τ respectively 2.

The effect of the error is two-fold. Firstly, the constraints will be violated and so the

energy swap is not complete. In general, the quantum fluctuation will experience a two-

mode squeezing, i.e., the annihilation operator transforms as a1 → A1a1+A2a†1+B1a2+

B2a†23 . Let us assume that the qubit is initially a thermal state with motional excitation

〈n1〉in, the erroneous SBS will yield a final excitation of quantum fluctuation as 〈n1〉q =(|A1|2 + |A2|2)〈n1〉in + |A2|2 + |B2|2. Secondly, the error potential will cause unexpected

acceleration on the ions, so the final classical position and momentum will be uncertain,

and hence will lead to motional heating. A position discrepancy δx1 and a momentum

discrepancy δp1 will give an excitation of classical motion as 〈n1〉c = (δx1)2(m1/ν0/2~)+

(δp1)2/(2~m1ν0). The total motional excitation after the process will be 〈n1〉c + 〈n1〉q.

I have numerically assess the magnitude of motional heating caused by such Gaussian

potential fluctuation with different κ and τ . The result is shown in Fig. 8.6. In all cases

investigated, the motional excitation due to incomplete energy swap is insignificant when

comparing to that due to the uncertainty of classical displacement. Nevertheless, precise

manipulation of ions’ classical position and momentum has been recently demonstrated

in ion shuttling and splitting experiments [175, 33]. So I claim that, by using such

experimental techniques, the SBS cooling scheme can be realised without causing serious

motional heating. I further note that the actual motional excitation due to the random

potential error is highly dependent on the experimental implementation of the global

potential. The model of global potential in Eq. (8.17) is far from optimal, because a

strong potential has to be applied (applying high voltage to the electrodes) to implement

the displaced harmonic well when the ions are far separated. The same situation can be

implemented with much weaker potential by using segmented traps, which can reduce

the magnitude of the random error.

8.10 Conclusion

In this chapter, I propose that the axial motional excitation of an ion qubit can be

removed by a swapping beam splitter. The SBS can be implemented by a controlled

2In the numerical simulation, I added a random error, which follows a Gaussian distribution withwidth κ, to the potential at every time interval τ . The potential error between two time intervals isinterpolated with respect to time.

3In the ideal case that energy transfer is complete, the parameters are A1 = A2 = B2 = 0 andB1 = eiθ for some real angle θ.

140


Figure 8.6: (Left) Motional excitation caused by the uncertainty of classical displacement,〈n1〉c. The spread of the Gaussian error is κ = 10−3 (solid) and κ = 10−4 (dashed).(Right) Parameters that determines the motional excitation due to incomplete phononswap. |A1|2 + |A2|2 (red) and |A2|2 + |B2|2 (green) is plotted for the case with κ = 10−3

(solid) and κ = 10−4 (dashed). These results show that 〈n1〉q is comparable to 〈n1〉c onlyif 〈n〉in is at the order of 105, which is unconventionally high in trapped ion quantumlogic experiments.

collision between the qubit and a ground state coolant ion, i.e., the ion separation is

precisely controlled that the mutual Coulomb interaction swaps the motional state of the

ions. The whole process can take less than ten oscillation periods of the trap, which is at

µs range for current state MHz traps. The cooled individual ions can then be diabatically

combined into a single well for high fidelity quantum logic operation. I have outlined a

systematic procedure to obtain the time variation of the trap parameters for both the

SBS and the ion combination processes. If excessive coolants are prepared before the

quantum computation, a new coolant can be employed in each round of cooling, thus

lengthy laser cooling is not necessary during the computation. Therefore, the cooling

scheme can improve the operational speed of an ion trap QC.

I note that the core of my SBS cooling scheme is the tuneable trap frequency and

quadratic interaction of two coupled harmonic oscillators. The idea could also be applied

to cool systems with similar behaviours, such as polar molecules [87] and nanomechanical

oscillators [82, 166].

141

Chapter 9

Summary

Aiming to improve the practicality of quantum information processing (QIP), in my

PhD period I have studied various applications in quantum cryptography and ion trap

quantum computation.

In the work presented in Ch. 2, we have examined the assumptions behind a new

quantum cryptography scheme called position-based quantum cryptography (PBQC).

We discovered that an incorrect assumption was made in the previous literature that

the cheaters do not share entangled resources. In fact, we showed that all known PBQC

protocols could be deterministically cheated if the cheaters share entanglement. By

generalising the entanglement attack in our work, the most general PBQC protocol was

shown to be insecure if the cheaters employ sufficient entangled resources to conduct

nonlocal measurement. Therefore PBQC is not unconditionally secure, even when all

apparatus is perfect. Although PBQC is still claimed to be secure in the condition that

the cheaters share only bounded amount of entanglement [40], conducting a useful PBQC

protocol requires demanding infrastructure, such as straight and concrete transmission

channels, and extremely fast detectors and transmitters. Based on these issues, PBQC

is unlikely a practical application of QIP.

Nevertheless, there are still academically interesting problems inspired by PBQC.

In all known cheating strategies today, the success rate scales as a polynomial of the

dimension of the encrypting quantum system. In principle, if the message is encrypted in

an infinite-level quantum system, successful cheating would require unrealistically large

amount of entangled resources. Such an infinite-level system can be found in continuous-

variable (CV) systems, which the information can be an arbitrary complex number.

However, according to an ongoing work that I am studying, if the message is encrypted

as a Gaussian state, then the cheaters can reproduce the correct response if they share

a EPR state and use it to conduct a CV version of the teleportation attack presented

142

Chapter 9. Summary

in Ch. 2. Therefore, employing an infinite-level system does not necessarily improve the

security of PBQC. A possibly more secure scheme can be a variation of Protocol A that

the message is encrypted as either a coherent state or a Fock state. Since the Fock basis is

not Gaussian, a more complicated entanglement attack is deemed required. I believe that

both the protocol and the corresponding cheating strategy of CV PBQC are interesting

directions for further investigations.

In the work presented in Ch. 3, we have analysed the security of quantum secret

sharing (QSS) when CV cluster states are employed as the resources. We proposed a

procedure to transform a multipartite QSS scheme to a bipartite quantum key distri-

bution (QKD) scheme, so the security of QSS can be analysed by using well-developed

techniques in CV QKD. While in the literature the security of CV QSS is guaranteed

only when the resources state is infinitely squeezed, our studies show that a finitely but

sufficiently squeezed CV cluster state can also produce a nonzero secret sharing rate.

Our results relax the stringent requirement of the resources that could be used in secure

CV QSS, thus the practicality of QSS is improved.

However, there are more problems have to be solved before QSS is practically useful.

One problem is that realistic apparatus is imperfect, for example the transmission chan-

nels are lossy, and the state initialisation and detection are not 100% accurate. These

imperfections may weaken the correlation between the dealer and the access structure,

thus reduce the secret sharing rate. Besides, our calculations are conducted in the asymp-

totic limit that infinite rounds of state distribution have been conducted; in practice secret

sharing rate has to be calculated with the finite-key effect considered. Nevertheless, by

using the transformation procedure in our work, the imperfection analysis in CV QKD

could be borrowed to study the security of QSS under practical situations.

In the work presented in Ch. 5, we studied how the information encoded in the ion

electronic states would be influenced by the electric field for transporting ion qubits.

We found that the electric field would produce a dc Stark shift on the energy levels,

which induces a phase shift on the quantum information. We obtained the optimal

transportation trajectory that minimises the accumulative dc Stark shift, and deduced

a threshold speed above which the phase shift would become a significant error on the

quantum information. Our calculation shows that the threshold speed is at least two

orders of magnitude faster than the transportation speed in state-of-the-art ion trap

experiments. Our work verifies previous claim that dc Stark effect is not a major source

of error in ion trap quantum computation.

As a by-product of this work, we have derived a formula, Eq. (5.3), to describe the

motional state transformation of an ion after being transported by a harmonic well. This

143

Chapter 9. Summary

formula is useful for deducing the optimal well trajectory that causes negligible motional

excitation after the transportation; such optimal transportation has been realised in

experiments [33]. A generalisation of this formula is to consider the transportation of a

chain of ions. According to our preliminary work shown in Appendix A.6, if all ions have

the same mass, Eq. (5.3) is still applicable to describe the centre-of-mass mode, while

other motional modes will not be excited if the harmonic well strength is kept constant.

However, the situation becomes much complicated if the ions have unequal mass. This is

because the motional modes are coupled, which the general analytic solution is difficult

to obtain. Nevertheless, it is still of high interest to develop a numerical procedure

to deduce the optimal trajectory for inhomogeneous ion chain transportation. Because

each harmonic well in recent experiments usually involves one qubit and one coolant ion,

a transportation process with lower motional excitation rate would reduce subsequent

cooling time.

In the work presented in Chs. 6 and 7, we proposed a new application of ion trap

systems: universal bosonic simulator (UBS). Being trapped in a harmonic potential, the

quantised ion motion exhibits bosonic behaviours, so each degree of freedom of ion motion

can simulate a bosonic mode. In the architecture in Ch. 6, which involves multiple ions

being trapped in a single harmonic well, boson initialisation and transformation can be

conducted by applying radiation fields with precisely tuned frequency. Although the

speed and the quality of the operations would be reduced when much more than four

modes are simulated, this architecture is implementable with today’s technology and is

useful to simulate small-scale bosonic phenomena, such as the Hong-Ou-Mandel effect.

In the UBS architecture in Ch. 7, which involves separately trapped ions, boson

initialisation and transformation can be conducted by varying the trap potential. This

architecture has the advantage that the operation applied on one mode would barely

influence the uninvolved modes, and both the speed and the quality of each operation

are independent of the total number of modes in the simulation. Implementing this

architecture requires the trap potential to be controlled with high speed and accuracy;

such level of control has been demonstrated in recent experiments [33, 175].

When comparing to conducting optical experiments, the ion trap UBS has the advan-

tage that a wider range of state can be deterministically initialised, stronger nonlinear in-

teraction can be applied, and higher efficiency measurements can be conducted. However,

the practicality of the UBS depends on the level of imperfections, such as the fluctuation

of trap potential that excites the motional states, the higher order non-harmonic poten-

tial that induces unwanted coupling of modes, and the infidelity of bosonic operations

that is due to the imprecision of the trap potential and of the applied radiation field.

144

Chapter 9. Summary

The magnitude of these imperfections has to be verified in experiments. Nevertheless, the

detrimental effects of the imperfections can be relieved by imposing rectifying methods.

For example, the speed of Fock state initialisation and measurement can be improved

by using composite pulses of radiation field [78], so the motional heating due to poten-

tial fluctuation can be reduced. Another method is to employ dynamical decoupling to

remove undesired interaction. Furthermore, when comparing to quantum computation,

less accuracy is generally required to produce meaningful results in quantum simulations

[41]. In summary, we believe UBS is a promising application of ion trap systems that

deserves further investigations.

In the work presented in Ch. 8, I proposed a new method to re-cool ion qubits during

quantum computation. The principle is to apply a swapping beam splitter to transfer

the motional excitation of the qubit to a coolant ion that was prepared in the motional

ground state. I showed that by using this method, ions can be re-cooled for over ten

times faster than by using laser cooling. Since ion qubits have to be frequently re-cooled

for implementing high fidelity logical operations, my method can improve the operational

speed of ion trap quantum computers.

One possible threat to the cooling scheme is the fluctuation of trap potential. My

investigation shows that the influence of fluctuation is sensitive to how the potential is

implemented, e.g. the distribution of the electrodes and the ansatz of potential variation;

the actual performance of the cooling scheme has to be verified in experiments. Never-

theless, in the inverse-engineering procedure discussed in Sec. 8.4, the ansatz that I have

discussed yields a diabatic process, i.e., a shortcut to adiabaticity, which is generally

believed to be robust against parameter fluctuations. Additionally, the constraints that

I have proposed in Sec. 8.4 are sufficient but not necessary conditions for implementing

a swapping beam splitter; in practice the constraints can be optimised for experimental

convenience.

In summary, in my PhD research I have resolved various hardware and software issues

in quantum cryptography and ion trap quantum computation. I believe my works have

brought quantum information processing a tiny step closer to practical application.

9.1 Prospects

I believe the practicality of quantum information processing would be continually im-

proved by the advancement of apparatus and theoretical studies; there are several direc-

tions that the development is particularly promising. For example, room temperature

quantum storage has been realised in a single defect centre of a semiconductor crystal for

145

Chapter 9. Summary

over 30 minutes [157]. It is anticipated that if the quantum information is encoded in the

decoherence-free subspace of multiple defect centres, the coherent storage time can be

further extended. Such techniques might be useful for implanting quantum certification

in valuable crystals, such as diamonds. Because only well-trained experts could deter-

mine the grade of a diamond (the 4C’s), ordinary customers and jewellers rate a diamond

solely by trusting the certification report, which can be easily faked. By implementing

the idea of quantum money on diamonds, i.e., the certifying information is encoded in

non-orthogonal states of multiple defect centres (see e.g. Refs. [179] and [16]), the buyers

and sellers could have more reliable information about the genuineness of the diamond.

Another promising direction of development is building hybrid quantum devices. Dif-

ferent implementations of QIP, such as photons, ion traps, and superconductors, have

respective strengths and weaknesses. For instance, the measurement of trapped ion qubits

is high in fidelity but low in speed; on the other hand, the quantum logical operations

of superconducting circuits are fast but the readout of information suffers from certain

limitations. Quantum information processing is potentially more practical by building a

hybrid quantum device that is composed of and could take advantage of multiple physical

systems [186]. Nevertheless, new problem is encountered that the interaction between

quantum systems, such as between trapped atoms and superconducting circuits, is weak

and is volatile against environmental noise. Such problem could be relievable by extend-

ing a recent proposal that creates entanglement by dissipation [94]. Analysis showed that

such entanglement creation process is more resistant to noise. Designing a hybrid device

to realise the dissipation-mediated entanglement creation would be a promising direction

of research.

In addition to constructing a quantum information processor that implements the

circuit elements of universal quantum computation [143], an alternative processor im-

plementation is to create a large entanglement state for conducting measurement-based

quantum computation (MBQC). In today’s MBQC model, the resources state is assumed

to be created by appropriately controlling the interaction between particles. However,

in some quantum systems, such as the electron spins in condensed matter systems, the

particle-particle interaction is barely controllable. Nevertheless, it is easy to see that

if the particle interaction is of the Ising type, a cluster state will be created occasion-

ally. To the best of my knowledge, there are only a few studies about the application of

such persistently and uncontrollably interacting system in MBQC. I anticipate that fur-

ther theoretical studies of the computational model and the error tolerance would allow

practical MBQC to be implemented in a wider range of quantum system.

Just like classical communication systems, while long (kilometer) range transmission

146

Chapter 9. Summary

facilities could be installed in foreseeable future, a major obstacle to the practicality of

quantum communication is the ‘last mile problem’, i.e., it is not easy to install quantum

channels to deliver secure quantum signals to every household. This last mile problem

can be solved if the security of the quantum signal is preserved after passing through

existing infrastructure. One idea is to transmit quantum signals through copper wires,

such as telephone wires and television cables, that have been installed in every modern

building. Unfortunately, according to my preliminary investigation, a domestic copper

wire with centimetre length would expose to environmental noise that is serious enough

to destroy the security of the QKD scheme, if the signal’s frequency is below 100 MHz.

Nevertheless, in a recent experiment entanglement distribution through a metre-long

coaxial cable has been demonstrated by using microwave frequency signal [152]. It would

be promising to develop quantum communication protocols that could be implemented

by such techniques, so that secure quantum signals could be transmitted through existing

household copper wire networks.

147

Appendix A

Appendix

A.1 Security of Modified Protocol

In Sec. 2.7, I have demonstrated that Protocol A′ cannot be cheated by the teleportation-

based strategy. Recently, all PBQC protocols are shown to be insecure if the cheaters

possess entanglement resources that scales polynomially as the accuracy, 1/ǫ, where ǫ

is the failure rate of cheating. However, it is still an open question that what is the

minimum amount of entanglement resources required to cheat a general PBQC protocol.

In the following, I will prove that Protocol A’ cannot be 100% successfully cheated if

the cheaters share only a pair of entangled qubit or qutrit. Therefore the cheaters must

possess at least two Bell pairs for successful cheating.

A.1.1 Security Against Attacks with One Entangled Qubit

Let us first consider the two-verifier case. Since any two qubit entangled state can be

created by teleportation through a Bell state, without loss of generality I consider the

cheaters share a Bell state in Eq. (2.3). It is easy to observe that the state has to be

measured at t = (d− l)/c, otherwise any measurement outcome obtained after this time

cannot help making the correct response. As the B2 cheater possesses only the qubit

from the Bell state, he can measure the qubit according to the basis information from

V2 verifier. This action is actually a remote state preparation [28], which leaves the Bell

state qubit of B1 cheater in either

| ↑〉 = g(θ, φ)|0〉+ h(θ, φ)|1〉 or | ↓〉 = h∗(θ, φ)|0〉 − g∗(θ, φ)|1〉 , (A.1)

148

Chapter A. Appendix

where g, h depend on the basis information received by B2 cheater. The appearance of

| ↑〉 and | ↓〉, with the probability 50% each, depends on the measurement outcomes of

B2 cheater. This information will immediately be sent to B1 cheater together with the

encryption basis information and the measurement basis of the Bell state qubit.

Suppose the encryption is in the Z-basis, i.e., θ = 0. Once received the encrypted

qubit, B1 cheater conducts a von Neumann measurement with basis

|M1〉 = α|0〉|0〉+ β|1〉|1〉, |M2〉 = β∗|0〉|0〉 − α∗|1〉|1〉|M3〉 = γ|0〉|1〉+ δ|1〉|0〉, |M4〉 = δ∗|0〉|1〉 − γ∗|1〉|0〉 , (A.2)

for some complex coefficients α, β, γ, δ. The first bracket belongs to the encrypted qubit

and the second one belongs to the teleported qubit. The above measurement basis states

are chosen that each component of |0〉 and |1〉 of the teleported qubit is not associated

with the superposition |0〉 or |1〉 of the encrypted qubit. Otherwise if a measurement

basis state, say |M1〉, contains terms like (|0〉 + |1〉)|1〉, the von Neumann operator of

|M1〉 will project the encrypted qubit to a superposition state that the cheaters cannot

distinguish its original identity.

Since B1 cheater knows nothing about the encryption basis, the measurement con-

ducted always has the same measurement basis as in Eq. (A.2). For general θ and φ, B1

cheater gets one of the four states before the measurement:

|ψ0〉| ↑〉, |ψ0〉| ↓〉, |ψ1〉| ↑〉, |ψ1〉| ↓〉 . (A.3)

An important observation here is that the cheaters are able to distinguish the encoded

qubit, only if each measurement basis state |Mi〉 that contains the components of either

only |ψ0〉| ↑〉 and |ψ1〉| ↓〉, or only |ψ0〉| ↓〉 and |ψ1〉| ↑〉. This statement can be refor-

mulated to say that each state in Eq. (A.3) is a superposition of at most two states in

Eq. (A.2).

When |ψ〉| ↑〉 is expanded, we have

|ψ0〉| ↑〉 = cosθ

2g(θ, φ)|0〉|0〉+ cos

θ

2h(θ, φ)|0〉|1〉

+ sinθ

2eiφg(θ, φ)|1〉|0〉+ sin

θ

2eiφh(θ, φ)|1〉|1〉 . (A.4)

Without loss of generality, I assume it is a superposition of |M1〉 and |M3〉. By comparing

149

Chapter A. Appendix

the component coefficients in Eqs. (A.2) and (A.4), the following relations are imposed:

cotθ

2e−iφ g

h=α

βand cot

θ

2e−iφh

g=γ

δ. (A.5)

Similarly, |ψ1〉| ↓〉 can be expanded as

|ψ1〉| ↓〉 = sinθ

2h∗(θ, φ)|0〉|0〉 − sin

θ

2g∗(θ, φ)|0〉|1〉

− cosθ

2eiφh∗(θ, φ)|1〉|0〉+ cos

θ

2eiφg∗(θ, φ)|1〉|1〉 . (A.6)

This state has to be a superposition of either |M1〉 and |M3〉 or |M2〉 and |M4〉, in order

to avoid unphysical result 〈ψ0 ↑ |ψ1 ↓〉 6= 0.

I first consider |ψ1 ↓〉 is a superposition of |M2〉 and |M4〉. By comparing the compo-

nent coefficients in Eq. (A.2) and (A.6), the following relations have to be satisfied:

cotθ

2e−iφ g

h= −α

βand cot

θ

2e−iφh

g= −γ

δ. (A.7)

Together with Eq. (A.5), either α = δ = 0 and g(θ, φ) = 0, or β = γ = 0 and h(θ, φ) = 0,

for any (θ, φ). These relations imply that B2 cheater always measures the Bell state

qubit in the Z basis, and B1 cheater measures both encrypted qubit and Bell state qubit

individually in Z basis. As a result, the measurement of B1 cheater can project, for

example, both |ψ00〉 and |ψ10〉 to |M2〉. Therefore, the cheaters cannot always distinguish|ψ0〉 and |ψ1〉 from the measurement outcomes.

Now I consider another case that |ψ1 ↓〉 is a superposition of |M1〉 and |M3〉. By

considering the coefficients of components in Eq. (A.2) and (A.6), the following relations

are imposed

tanθ

2e−iφh

∗

g∗=α

β, tan

θ

2e−iφ g

∗

h∗=γ

δ. (A.8)

Together with Eq. (A.5), we get the following relations,

|g|2|h|2 = tan2 θ

2,|h|2|g|2 = tan2 θ

2. (A.9)

These two relations can be satisfied at the same time only when θ = π/2. I note that

according to Eq. (2.23), θ = π/2 implies that cheating is possible if the encoding qubit is

encrypted in the X basis. This explains why the cheating strategy in Sec. 2.4 is possible.

By the above arguments, I can claim that in the two-verifier case, Protocol A′ cannot

be cheated with 100% successful probability if the cheaters share only a pair of entangled

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Chapter A. Appendix

qubit. In the case with N > 2 verifiers, I can prove by contradiction that Protocol A′

is secure if the cheaters share less than or equal to N − 1 Bell pairs. In this case, there

must be a cheater possesses only one Bell state qubit, say the Bn cheater possess a qubit

that entangles with Bm cheater. Since the encrypted qubit can be sent from any verifier,

let us consider the case that Vn verifier sends the qubit. We can further assume that Ui’s

are identity except Um. This case reduces to the two-verifier case that have discussed in

previous paragraph, which cannot be cheated with 100% successful probability.

A.1.2 Security Against Attacks with One Entangled Qutrit

Here I outline the proof of security of Protocol A′ against arbitrary attacks if the two

cheaters possess only a pair of maximally entangled qutrit in the two-verifier case. First

of all, I discuss the necessary properties of a maximally entangled d-level system (qudit)

if the cheating can succeed with certainty. Let us consider B2 cheater conducts a remote

state preparation by measuring the qudit pair in a d-level basis, then the qudit of B1

cheater will become, with equal probability, one of the eigenstates of the d-level basis.

The choice of the d-level basis depends on the information of the encryption. If the

encrypted qubit is an eigenstate of Z, I assume B1 cheater will receive a state in the set

|φ1〉, . . . , |φd〉. Let us define a vector ~|Φ〉 ≡ (|φ1〉 . . . |φn〉)T . If the encrypted qubit is an

eigenstate of n(θ, φ) · ~σ, B1 cheater’s qudit will be an element of the vector T (θ, φ) · ~|Φ〉,where T (θ, φ) is a d × d unitary matrix function of polar angles that is known to both

cheaters.

Let B1 cheater measures the encrypted qubit and the entangled qudit in a basis with

eigenstates |M1〉, . . . , |Md〉. In order to distinguish the identity of the encrypted qubit

after obtaining information from B2 cheater, each |Mi〉 should not contain components of

both |φj〉|0〉 and |φj〉|1〉 for any j. Let us define a selection matrix S(i) that is a diagonal

matrix, where S(i)jj = 1 if |Mi〉 contains the component of |φj〉|0〉; and S

(i)jj = 0 if |Mi〉

contains the component of |φj〉|1〉. In terms of S(i), |Mi〉 can be written as

|Mi〉 =d∑

j,j′=1

(

αij′S(i)j′j |φj〉|0〉+ αij′(I − S

(i)j′j)|φj〉|1〉

)

, (A.10)

where |αi1|2 + . . . + |αid|2 = 1; I is the d × d identity matrix. Since B1 knows nothing

about the basis, his measurement is always the same.

Similar to the argument above, if B1 cheater is able to distinguish between |ψ〉 and

151

Chapter A. Appendix

|ψ1〉 for general θ and φ, |Mi〉 has to be

|Mi〉 =d∑

k,k′,j=1

(

βikS(i)kk′Tk′j|φj〉|ψ0〉+ βik(I − S

(i)kk′)Tk′j |φj〉|ψ1〉

)

, (A.11)

where |βi1|2 + . . . + |βin|2 = 1; S(i) is the selection matrix for the specific θ and φ. By

comparing Eqs. (A.10) and (A.11), we have

αij[cosθ

2S(i)jj + sin

θ

2eiφ(I − S

(i)jj )] = βikS

(i)kk Tkj , (A.12)

αij [sinθ

2S(i)jj − cos

θ

2eiφ(I − S

(i)jj )] = βik(I − S

(i)kk )Tkj . (A.13)

After summing the above two relations, and taking the scalar product of themselves, we

have

d∑

j

αij [(1 + sin θ)S(i)jj + (1− sin θ)(I − S

(i)jj )]α

∗ij =

∑

k

βikβ∗ik = 1 , (A.14)

where the identities S2 = S, (I − S)2 = (I − S), and S(I − S) = 0 are employed; the

relation on the right hand side is the normalisation condition of Eq. (A.11). The above

equation is true for any θ except when sin θ 6= 0, so the following relation can be deduced:

d∑

j

αijS(i)jj α

∗ij =

∑

j

αij(I − S(i)jj )α

∗ij =

1

2, (A.15)

where I have implicitly employed the normalization condition of |Mi〉 in Eq. (A.10). This

relation restricts the kind of measurement that should be conducted by B1 cheater, if the

cheating is successful with 100% probability.

In the following, I will show that there does not exist a set of |Mi〉 for d = 3

that Eq. (A.15) is satisfied. Let us assume that at least one measurement basis state, say

|M1〉, consists of three components, say |φ1〉|0〉, |φ2〉|0〉, |φ3〉|1〉, with nonzero coefficient.

Then |M2〉, . . . , |M6〉 should consist of more than one component that is the same as |M1〉.Let us consider |M2〉 shares two common components as |M1〉; for example |M2〉 consistsof |φ1〉|0〉, |φ2〉|1〉, |φ3〉|1〉. By the completeness of the measurement basis states, there

must be a state, say |M3〉, that consists of |φ1〉|1〉. However, such a state cannot be

orthogonal to both |M1〉 and |M2〉 while satisfying Eq. (A.10). This is because the |φ3〉component of |M3〉 should have either finite or no overlap with both |M1〉 and |M2〉,while the |φ2〉 component of |M3〉 should overlap only with either |M1〉 and |M2〉. As

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Chapter A. Appendix

a consequence |M2〉 must contain |1〉|φ1〉, |1〉|φ2〉, |0〉|φ3〉. By applying the same idea

to |M3〉, . . . , |M6〉, at least three of the six measurement basis states must consist of the

same set of components.

Let us consider |M1〉, |M3〉, |M5〉 consist of |0〉|φ1〉, |0〉|φ2〉, |1〉|φ3〉. With Eq. (A.15)

respected, the states should be expressed as

|Mi〉 =1√2

(

cos θi|0〉|φ1〉+ sin θieiµi |0〉|φ2〉+ eiνi|1〉|φ3〉

)

. (A.16)

Since the three states are orthogonal, we require

cos θi cos θj + sin θi sin θjei(µi−µj) = −ei(νi−νj) , (A.17)

for i 6= j. The norm of the term on the right hand side is equal to 1; the norm of the left

hand side term is equal to 1 if and only if θi = θj , µ1 = µj, and νi = νj + π. In terms of

the parameters of |M1〉, |M3〉 can be expressed as

|M3〉 =1√2

(

cos θ1|0〉|φ1〉+ sin θ1eiµ1 |0〉|φ2〉 − eiν1|1〉|φ3〉

)

. (A.18)

However, there does not exists a set of (θ5, µ5, ν5) that both satisfies Eq. (A.17) and makes

|M5〉 orthogonal to |M1〉 and |M3〉. Therefore, there does not exist three-component

measurement basis states that all the above criteria are satisfied. In other words, Protocol

A′ cannot be cheated if B1 cheater measures the encrypted qubit and the qutrit in a basis

with three-component states.

I have assumed above that at least one measurement basis state is a superposition

of three components. I now consider all states contain only two components. With

Eq. (A.15) satisfied, the states are given by

|M1,2〉 =1√2

(

|0〉|φ1〉 ± eiµ1 |1〉|φ2〉)

(A.19)

=1√2|ψ0〉

(

cosθ

2|φ1〉 ± sin

θ

2ei(µ1−φ)|φ2〉

)

+1√2|ψ1〉

(

sinθ

2|φ1〉 ∓ cos

θ

2ei(µ1−φ)|φ2〉

)

,

|M3,4〉 =1√2

(

|0〉|φ2〉 ± eiµ2 |1〉|φ3〉)

(A.20)

=1√2|ψ0〉

(

cosθ

2|φ2〉 ± sin

θ

2ei(µ2−φ)|φ3〉

)

+1√2|ψ1〉

(

sinθ

2|φ2〉 ∓ cos

θ

2ei(µ2−φ)|φ3〉

)

,

|M5,6〉 =1√2

(

|0〉|φ3〉 ± eiµ3 |1〉|φ1〉)

(A.21)

=1√2|ψ0〉

(

cosθ

2|φ3〉 ± sin

θ

2ei(µ3−φ)|φ1〉

)

+1√2|ψ1〉

(

sinθ

2|φ3〉 ∓ cos

θ

2ei(µ3−φ)|φ1〉

)

,

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Chapter A. Appendix

or some cyclic permutation of |φi〉’s. On the other hand, because I have proved every

|Mi〉 cannot contain three components, they should be written as,

|M1,2〉 =1√2

(

|ψ0〉(

T1i|φi〉)

± eiν1|ψ1〉(

T2i|φi〉)

)

, (A.22)

|M3,4〉 =1√2

(

|ψ0〉(

T2i|φi〉)

± eiν1|ψ1〉(

T3i|φi〉))

, (A.23)

|M5,6〉 =1√2

(

|ψ0〉(

T3i|φi〉)

± eiν1|ψ1〉(

T1i|φi〉)

)

. (A.24)

Let us consider the qutrit state associated with |ψ1〉 in |M1〉, and that associated with

|ψ0〉 |M3〉. Although both states should be(

T2i|φi〉)

in Eq. (A.22) and Eq. (A.23), but

they are unequal in Eq. (A.19) and Eq. (A.20). Therefore, there does not exist two-

component measurement basis states that all the above criteria are satisfied. As a result,

Protocol A′ cannot be cheated with 100% successful probability in the two-verifier case

if the cheaters only share one pair of maximally entangled qutrit. I note that, by using

similar argument as in Sec. A.1.1, N -verifier Protocol A′ cannot be cheated with 100%

successful probability if the cheaters share less than N qutrits.

A.2 Example of CC QSS

A.2.1 Example 1: (2,3)-CC protocol

In a (2,3)-CC protocol, the access structure is any two of the three parties collaborating,

while the adversary structure is any collaboration with only one party. The (2,3)-CC

protocol can be implemented by a linear three-mode cluster, as shown in Fig. A.1. I

assume the dealer picks the secret classical value s according to a Gaussian probability

distribution with a width Σ, i.e.,

PD(s) =1√πΣ

e−s2/Σ2

. (A.25)

The state is encoded by displacing mode 2 by is/√2 and mode 3 by −is/

√2. In the

infinitely squeezed case, the nullifiers of the cluster state are

N1 = p1 − q2 − q3 ; N2 = p2 − q1 + s ; N3 = p3 − q1 − s . (A.26)

154

Chapter A. Appendix

1

2 3

1 1

1

5 2

1 1

1

4 3

1

1

Figure A.1: Schematic representation of the cluster states for the (2,3)-protocol (left),and the (3,5)-protocol (right). Each oval represents a squeezed mode, which will bedistributed to the party denoted by the label inside. The subscript of each mode denotesthe squeezing parameter and the displacement before the CPHASE operation (for clarityof the graph, each displacement is divided by i/

√2). The edges joining the modes

represent CPHASE operations, of which the strength is denoted by the edge’s label.

A.2.1.1 Parties 1,2 collaboration

Let us consider parties 1 and 2 to be the access structure. If the cluster state is infinitely

squeezed, the shared secret is the difference between p measurement outcome of party 1

and q measurement outcome of party 2, i.e., s = q1 − p2 according to N2. Derived from

Eq. (3.8), the reduced Wigner function of party 3 is a constant function independent of

s, thus the protocol is secure.

In the finitely squeezed case, the Wigner function of the cluster state is given by

Eq. (3.7) with the nullifiers in Eq. (A.26). For simplicity, I assume the modes are equally

squeezed, i.e., σi = σ for all i, but my analysis is applicable to the states with inhomoge-

neous σi. The measurement basis of the parties is the same as in the infinitely squeezing

case. The classical probability of having the measurement outcomes p2 and q1 is

P(s)A|D;1,2(q1, p2) =

1

πe−(p2−q1−s)2/σ2

e−σ2q21 , (A.27)

which is obtained by tracing out q1, p2, and the contributions of party 3 in the Wigner

function. The probability distribution of the difference of the outcomes, s′ = q1−p2, can

be obtained by tracing out an orthogonal quantity, e.g. (p1 + q2)/2. Then we get

PA|D;1,2(s, s′) =

1√πσ

exp(

− (s− s′)2

σ2

)

, (A.28)

155

Chapter A. Appendix

and thus according to Eq. (3.12),

PA;1,2(s′) =

1√π√σ2 + Σ2

e−s′2/(σ2+Σ2) . (A.29)

The mutual information between the dealer and the access structure can then be calcu-

lated using Eq. (3.13).

I now consider the adversary structure. The reduced Wigner function of ρE and

ρE|D(s) are

WE|D;3 =σ2

π√1 + σ4

e−σ2((p3−s)2+q23(1+σ4))

1+σ4 ; (A.30)

WE;3 =

∫

WE|D;3ds =σ2

π√1 + σ4 + σ2Σ2

e−σ2

(

q23+p23

1+σ4+σ2Σ2

)

. (A.31)

The covariance matrices of these states are given by

VE|D;3 =

(

12σ2 0

0 1+σ4

2σ2

)

, VE;3 =

(

12σ2 0

0 1+σ4+σ2Σ2

2σ2

)

, (A.32)

where the symplectic eigenvalues are νE|D;3 =√1 + σ42σ2 and νE;3 =

√1 + σ4 + σ2Σ2/2σ2,

respectively. The Holevo bound can then be calculated using Eq. (3.20) and (3.22), and

hence the secret sharing rate can be obtained from Eq. (3.23).

Because both parties 2 and 3 hold the end mode of the cluster state, their states

are local-unitarily equivalent, i.e., all Wigner functions will be the same as above except

replacing the subscript 2 by 3 and every value s by −s. The security for party 1, 3collaboration can be analysed by the same procedure as the 1, 2 collaboration, and the

secret sharing rate of both collaborations will be the same.


Let us consider now that parties 2 and 3 are the access structure. In the infinitely squeezed

case, because the operator N2 − N3 = p2 − p3 + 2s is also a nullifier, the secret s can be

obtained if both parties conduct p measurement, i.e., s = (−p2 + p3)/2. The protocol is

secure because the reduced Wigner function of party 1 is a constant independent of s.

In the finitely squeezed case, the measurement outcomes of p1 and p2 follow the

probability distribution

P(s)A|D;2,3(p1, p2) =

1

π√2 + σ4

e− (p2−p3+2s)2

σ2(2+σ4) e−σ2((p2+s)2+(p3−s)2)/(2+σ4) . (A.33)

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Chapter A. Appendix

The first exponent accounts for the strong correlations while the last exponent is re-

sponsible for higher order weak correlations. The quantity s′ = (−p2 + p3)/2 follows the

probability distribution

PA|D;2,3(s, s′) =

1√2πσ

exp(

− 2(s− s′)2

σ2

)

, (A.34)

and thus

PA;2,3(s′) =

√

2

π(σ2 + 2Σ2)exp

(

− 2s′2

σ2 + 2Σ2

)

. (A.35)

For the adversary structure, party 1, the Wigner function of the reduced state ρE|D;1

is given by

WE|D;1 =σ2

π√2 + σ4

exp

(

−σ2(

q21 +p21

2 + σ4

)

)

. (A.36)

Because Eq. (A.36) is independent of s, the Wigner function of ρE;1 and ρE|D;1 would

be the same, i.e., WE|D;1 = WE;1. Therefore the Holevo bound vanishes, i.e., party 1

cannot get any information, and hence the secret sharing rate is simply I(D : A).

A.2.2 Example 2: (3,5)-CC protocol

In a (3,5)-CC protocol, the access structure is any three of the five parties collaborating,

while the adversary structure is any collaboration with less than three parties. The

(3,5)-CC protocol can be implemented by a star-shaped five mode cluster, as shown in

Fig. A.1. All five modes of the cluster state are displaced by −is/√2, where the classical

secret s is assumed to be chosen according to the probability distribution in Eq. (A.25).

In the infinitely squeezed case, the nullifiers of the cluster state are given by

Ni = pi − qi+1 − qi−1 − s , (A.37)

where i+ 1 = 1 when i = 5; i− 1 = 5 when i = 1.

Ten different combinations of access structure can be formed in this protocol, but they

can be categorised into two classes of collaborations: three neighbouring parties, and two

neighbours with one disjoint party. Without loss of generality, I consider parties 1,2,3as an example of the three-neighbour collaboration, and parties 1,3,4 for two-neighbourcollaboration. The security analysis and secret sharing rate of these two examples can

be adapted to other collaborations after indices changing.

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Chapter A. Appendix

A.2.2.1 Parties 1,2,3 collaboration

Let us first consider parties 1, 2, and 3 are the access structure. In the infinitely squeezed

case, the secret s can be obtained if both parties 1 and 3 measure q and party 2 measures

p, i.e., s = −q1 + p2 − q3 according to N2 in Eq. (A.37). The reduced Wigner function of

parties 4, 5 collaboration is a constant function after tracing out the contributions of

the access structure in Eq. (3.8).

In the finitely squeezed case, the outcome probability of the measurement by parties

1,2,3 is determined by the reduced Wigner function WA|D;1,2,3(q1, p1, q2, p2, q3, p3),

which is obtained by tracing out the contributions of parties 4 and 5 in the full Wigner

function (Eq. (3.7) with the nullifiers in Eq. (A.37)). The measurement bases are q2, p1,

and q3; the outcome probability is obtained by tracing out the dependence of p1, q2, and

p3 from the reduced Wigner function. The probability distribution of the received secret,

s′ = p2 − q1 − q3, can be obtained by first substituting the set of variables (q1, p2, q3) by

another linearly independent set of variables, e.g., (q1, s′, q3), and then tracing out the

independent variables, i.e., q1 and q3. The Jacobian matrix of this variable transformation

is 1, so the form of probability distribution remains the same [13].

Each process of trace-out described above involves a specific physical meanings, but

the end result is that all contributions except s′ are traced out. So the probability

distribution of s′ can be obtained in only two steps: first, substituting one variable with

s′, e.g. p2 = s′ + q1 + q3, in the full Wigner function, and then tracing out all variables

except s′. In this case, we get

PA|D;1,2,3(s, s′) =

1√πσ

exp(

− (s− s′)2

σ2

)

, (A.38)

and thus

PA;1,2,3(s′) =

1√π√σ2 + Σ2

e−s′2/(σ2+Σ2) . (A.39)

For the adversary structure parties 4 and 5, the reduced Wigner function WE|D;4,5

and WE;4,5 can be obtained by tracing out the contribution of parties 1,2,3. The co-

158

Chapter A. Appendix

variance matrices of these states are

VE|D;4,5 =

12σ2 0 0 1

2σ2

0 1σ2 +

σ2

21

2σ2 0

0 12σ2

12σ2 0

12σ2 0 0 1

σ2 +σ2

2

, (A.40)

VE;4,5 =

12σ2 0 0 1

2σ2

0 1σ2 +

σ2+Σ2

21

2σ2 0

0 12σ2

12σ2 0

12σ2 0 0 1

σ2 +σ2+Σ2

2

, (A.41)

where the symplectic spectrum are νE|D;4,5 = √1 + σ4/2σ2,

√1 + σ4/2σ2 and νE;4,5 =

√1 + σ4/2σ2,

√1 + σ4 + 2σ2Σ2/2σ2. The von Neumann entropy can then be calculated

by using Eq. (3.20), and the secret sharing rate is calculated from Eq. (3.23).


Let us consider parties 1, 3, and 4 are the access structure. In the infinitely squeezed

case, the secret s can be obtained when party 1 measures p, party 3 and 4 measures

p′ = (p − q)/√2. Because N1 − N3 − N4 is a nullifier, their measurement results are

correlated as −p1 +√2p′3 +

√2p′4 = s. The reduced Wigner function of parties 2,5

collaboration is a constant function, so the secret sharing is secure.

In the finitely squeezed case, similar to the case of 1,2,3 collaboration, the prob-

ability distribution of the quantity s′ = −p1 + p3 − q3 + p4 − q4 can be obtained by

first substituting the set of variables (q1, p1, q3, p3, q4, p4) with the new set of variables

(q1, s′, q3, p3, q4, p4) in the Wigner function. The determinant of the Jacobian matrix of

this transformation is 1. All variables except s′ are traced out from the full Wigner

function, then we get

PA|D;1,3,4(s, s′) =

1√3πσ

exp(

− (s− s′)2

3σ2

)

, (A.42)

and thus

PA;1,3,4(s′) =

1√π√3σ2 + Σ2

e−s′2/(3σ2+Σ2) . (A.43)

For the adversary structure parties 2 and 5, the covariance matrices of the states

159

Chapter A. Appendix

Figure A.2: Secret sharing rate of CC QSS protocols using CV cluster states withdifferent squeezing parameters σ. The variance of the classical secret probability is chosenas Σ = 1. Left panel: (2,3)-protocol with 2,3 collaboration (solid line) and 1,3collaboration (dashed line) as the access structure. Left panel: (3,5)-protocol with 1,2,3collaboration (solid line) and 1,3,4 collaboration (dashed line) as the access structure.

ρE|D;2,5 and ρE;2,5 are

VE|D;2,5 =

12σ2 0 0 0

0 1σ2 +

σ2

20 1

2σ2

0 0 12σ2 0

0 12σ2 0 1

σ2 +σ2

2

, (A.44)

VE;2,5 =

12σ2 0 0 0

0 1σ2 +

σ2

2+ Σ2

20 1

2σ2 +Σ2

2

0 0 12σ2 0

0 12σ2 +

Σ2

20 1

σ2 +σ2

2+ Σ2

2

. (A.45)

The symplectic spectrum are νE|D;2,5 = √1 + σ4/2σ2,

√3 + σ4/2σ2 and νE;2,5 =

√1 + σ4/2σ2,

√3 + σ4 + 2σ2Σ2/2σ2, respectively. The von Neumann entropy can then

be calculated by Eq. (3.20), and hence the secret sharing rate by Eq. (3.23).

The secret sharing rate for the (2,3)- and (3,5)-protocols is plotted against σ in

Fig A.2. Apart from the 2, 3 collaboration in (2,3)-protocol that the correlation can be

completely removed from the adversary structure, CC QSS is secure unless the squeezing

parameter is larger than some threshold limit.

160

Chapter A. Appendix

σ

σ1

σ2 3

σ

σ1

σ5 2

σ σ4 3

σd

D

σdD

-1

Figure A.3: Schematic representation of the cluster state for the (2,3)-CQ protocol (left),and the (3,5)-CQ protocol (right). All of the modes have zero displacement before theCPHASE operation. The strength of unlabelled edges is A = 1.

A.3 Examples of CQ QSS

A.3.1 Example 1: (2,3)-CQ protocol

In a (2,3)-CQ protocol, any collaboration involving two of the three parties can form a

strong correlation with the dealer, while any one party alone is only weakly correlated

with the dealer. The protocol can be implemented by a diamond-shaped CV cluster state

with AD3 = A13 = A12 = 1 and AD2 = −1, as shown in Fig. A.3. 1 In the infinitely

squeezed case, the nullifiers are

ND = pD + q2 − q3 ; N1 = p1 − q2 − q3

N2 = p2 + qD − q1 ; N3 = p3 − qD − q1 . (A.46)

The finitely squeezed state is described by Eq. (3.7) with the above nullifiers.

The access structure can be composed by parties 1, 2, 1, 3, and 2, 3. The state

possessed by collaborations 1, 2 and 1, 3 are equivalent up to a local unitary, because

the nullifiers of 1, 2 will be the same as that of 1, 3 if the dealer applies a π-phase

operation, F (π), to his mode. On the other hand, the collaboration 2, 3 possesses a

different state.

1The diamond-shaped CV cluster state is the same as the error filtration code in cluster state quantumcomputation [173].

161

Chapter A. Appendix


If parties 1, 2 are the access structure, the strong correlations are specified by the

nullifiers

ND − N1 = pD − p1 + 2q2 and N2 = qD − q1 + p2 . (A.47)

A global operation is applied on the access structure’s modes to transfer the strong

correlation to mode 2, i.e., mode 2 is treated as mode h. The transformation UA can

be implemented by various sequence of operations, but the final measurement results

and the covariance matrix are not affected. One possible choice is the 1,2 Decoding

Sequence: (i) apply exp(−iq1q2); (ii) then exp(ip1p2); (iii) finally F2(π).

After tracing out the modes other than mode D and mode 2, the covariance matrix

of the resultant state ρDA is given by

VDA;1,2 =

12σ2

D0 0 1

2σ2D

0 1σ2 +

σ2D

21σ2 0

0 1σ2

1σ2 +

σ2

20

12σ2

D0 0 σ2

2+ 1

2σ2D

. (A.48)

The covariance matrix will be revealed in the parameter-estimation stage when half of

the states are measured.

As VDA;1,2 is not in the standard form, i.e., Eq. (3.25), rectifying quantum operations

are applied onto the residual states. First of all, the variance of qD and pD are balanced

by squeezing mode D with the squeezing parameter

γD =

√

σDσ

√

2 + σ2σ2D . (A.49)

Next, mode 2 is squeezed to balance the coherent (off-diagonal) terms, i.e., 〈∆qD∆p2〉and 〈∆pD∆q2〉. The squeezing parameter is given by

γ2 =

√

2σD

σ√

2 + σ2σ2D

. (A.50)

In practice, both γD and γ2 can be obtained empirically from the results in parameter-

estimation stage, i.e., without knowing the squeezing parameter of the initial cluster

state. This squeezing stage will transform the state ρDA as

ρDA → ρ′DA = SD(γD)S2(γ2)ρDAS†2(γ2)S

†D(γD) , (A.51)

162

Chapter A. Appendix

where the covariance matrix becomes

V ′DA;1,2 =

V(2,3) 0 0 c

0 V(2,3) c 0

0 c V ′q 0

c 0 0 V ′p

. (A.52)

V(2,3) =√

2 + σ2σ2D/2σσD ; c = 1/

√2σσD; the variances of mode 2 are

V ′q =

1 + σ2σ2D

σσD√

2 + σ2σ2D

; V ′p =

(2 + σ4)√

2 + σ2σ2D

4σσD. (A.53)

I note that γD and V(2,3) are the same for any collaboration in the (2,3)-protocol,

because mode D is kept with the dealer that is not affected by operations on delivered

modes. The disparity between V ′q and V ′

p implies imbalanced noise for the quadratures q2

and p2, which has to be rectified by state-averaging. The state will then be transformed

as

ρ′DA → ρ′′DA =1

2FD(−π/2)ρ′DAF

†D(−π/2) +

1

2F2(−π/2)ρ′DAF

†2 (−π/2) . (A.54)

The covariance matrix of ρ′′DA is given by

V ′′DA;1,2 =

V(2,3) 0 c 0

0 V(2,3) 0 −cc 0 VA;1,2 0

0 −c 0 VA;1,2

, (A.55)

where VA;1,2 = (V ′q + V ′

p)/2. Finally, the modes are measured by the dealer and party 2

in either the q or p basis. The variances of the measurement outcomes will be given by

Eq. (A.55).

I note that ρ′′DA is not a Gaussian state because the stage-averaging process in

Eq. (A.54) is not a Gaussian operation. To calculate the secret sharing rate by us-

ing the techniques in Sec. 3.4.2, I can pretend the measurement results originates from a

Gaussian state ρG where its covariance matrix is V ′′DA;1,2. According to Refs. [184, 68],

Gaussian states minimise the secure key rate for all states with the same covariance

matrix. Therefore state-averaging maximises the power of the unauthorised parties and

lower-bounds the secret sharing rate.

By comparing Eqs. (A.55) and (3.25), the variance of the dealer’s mode is V = V(2,3),

163

Chapter A. Appendix

and the analogous channel parameters can be deduced as τ = c2/(V 2(2,3) − 1/4) = 1

and χ = VA;1,2 − V(2,3). The minimal secret sharing rate can then be calculated by

Eqs. (3.31)-(3.34) and (3.36).


If parties 2, 3 are the access structure, the strong correlations are specified by the

nullifiers

ND = pD + q2 − q3 andN2 − N3

2= qD +

p22

− p32. (A.56)

The quantum correlations can be transferred to mode 2 by the 2,3 Decoding Sequence,

which is simply a 50:50 beam splitter that transforms q2 → −(q2 + q3)/√2, p2 → −(p2 +

p3)/√2, q3 → (q2 − q3)/

√2, and p3 → (p2 − p3)/

√2. The resultant covariance matrix of

mode D and mode 2 becomes

VDA;2,3 =

12σ2

D0 0 1√

2σ2D

0 1σ2 +

σ2D

21√2σ2 0

0 1√2σ2

12σ2 0

1√2σ2

D

0 0 σ2

2+ 1

σ2D

. (A.57)

The secret sharing rate can be deduced by similar processes as in the 1, 2 collab-

oration: squeezing and transforming local modes to construct a state with standardised

covariance matrix, and then measuring the states to obtain the analogous channel param-

eters for computing the information of different parties. However, the 2, 3 collaborationis special as the dealer and party 2 are actually holding a pure state, i.e., the beam split-

ter has removed all entanglement from the unauthorised parties. This can be seen from

the symplectic spectrum of Eq. (A.57), ν2,3 = 1/2, 1/2, so the entropy of the system

DA vanishes, i.e., S(DA) = 0. Therefore the unauthorised parties cannot obtain any

information about the secret by entangling their modes to the dealer’s mode. The secret

sharing rate is hence the same as the mutual information between the dealer and the

access structure, which is given by

I(D : A) = log(2V(2,3)) . (A.58)

A.3.2 Example 2: (3,5)-CQ protocol

In a (3,5)-CQ protocol, any collaboration with three of the five parties can form a strong

correlation with the dealer, while any collaboration with less than two parties is only

164

Chapter A. Appendix

weakly correlated with the dealer. The protocol can be implemented by a pentagonal

CV cluster state, as shown in Fig. A.3, where each connected vertex is entangled by a

CPHASE operation with Aij = 1. In the infinitely squeezed case, the nullifiers are

ND = pD −5∑

i=1

qi ; Ni = pi − qi+1 − qi−1 − qD , (A.59)

where i+1 = 1 when i = 5; i−1 = 5 when i = 1. The finitely squeezed state is described

by the Wigner function in Eq. (3.7) with the above nullifiers.

The access structure can be composed by two categories of collaboration: three neigh-

bouring parties, e.g. parties 1, 2, 3, and two neighbours with one disjoint party, e.g.

1, 3, 4. The collaborations in each category hold nullifiers with the same form, so the

decoding sequence will be the same. If the squeezing parameter is identical for all five

modes, the secret sharing rate of the collaborations in each category will also be the

same.


If parties 1, 2, 3 are the access structure, the strong correlations are specified by the

nullifiers

ND − N1 + 2N2 − N3 = pD − (p1 + 3q1) + (2p2 + q2)− (p3 + 3q3)

and − N2 = qD − p2 + q1 + q3 . (A.60)

The quantum correlations can be transferred to party 2 by the 1,2,3 Decoding Sequence:(i) apply exp(−iq1q2) and exp(−iq2q3); (ii) then exp(ip1p2) and exp(ip2p3); (iii) finally

exp(ip2(2p1 + q1)) and exp(ip2(2p3 + q3)).

The covariance matrix of the state ρDA of mode D and mode 2 is given by

VDA;1,2,3 =

12σ2

D0 0 1

2σ2D

05+σ2σ2

D

2σ25

2σ2 0

0 52σ2

5+6σ4

2σ2 −σ2

12σ2

D0 −σ2 1+σ2σ2

D

2σ2D

. (A.61)

All terms in VDA;1,2,3 can be revealed by x and p measurements in the parameter-

estimation stage except for the local coherent terms 〈(∆q2∆p2 + ∆p2∆q2)/2〉. However

these terms do not affect the parameters in the squeezing stage, and will be eventually

cancelled during state-averaging.

165

Chapter A. Appendix

The unmeasured states are locally squeezed to balance the variances of qD and pD,

as well as the coherent terms. The state is transformed as in Eq. (A.51), where the

parameters for the 1, 2, 3 collaborations are

γD =

√

σDσ

√

5 + σ2σ2D ; γ2 =

√

σ

5σD

√

5 + σ2σ2D . (A.62)

The covariance matrix of the transformed state is given by

V ′DA;1,2,3 =

V(3,5) 0 0 c

0 V(3,5) c 0

0 c V ′q 0

c 0 0 V ′p

(A.63)

where V(3,5) =√

5 + σ2σ2D/2σσD ; c =

√5/2σσD, and the variance of mode 2 is given by

V ′q =

(5 + 6σ4)√

5 + σ2σ2D

10σσD; V ′

q =5(1 + σ2σ2

D)

2σσD√

5 + σ2σ2D

. (A.64)

The value of γD and V(3,5) are the same for any collaboration in the (3,5)-protocol.

state-averaging ensues to balance the correlations of pD − q2 and qD − p2. Half of the

unmeasured states are transformed by FD(−π/2), while the other half are transformed

by F2(−π/2). After discarding the choice of division, the state transforms as Eq. (A.54),

and the covariance matrix becomes

V ′′DA;1,2,3 =

V(3,5) 0 c 0

0 V(3,5) 0 −cc 0 VA;1,2,3 0

0 −c 0 VA;1,2,3

, (A.65)

for VA;1,2,3 = (V ′q + V ′

p)/2 with the definition in Eq. (A.64). I note that the local

coherent terms vanish after state-averaging because their sign in FD(−π/2)ρ′DAF†D(−π/2)

and F2(−π/2)ρ′DAF†2 (−π/2) are opposite.

As discussed before, the measurement results can be pretended as coming from a

Gaussian state with the same covariance matrix V ′′DA;1,2,3. The variance of the dealer’s

mode is recognised as V = V(3,5), and the analogous channel parameters can be deduced

as τ = c2/(V 2(3,5) − 1/4) = 1 and χ = VA;1,2,3 − V(3,5). The minimal secret sharing rate

can then be calculated by Eqs. (3.31)-(3.34) and (3.36).

166

Chapter A. Appendix


If parties 1, 3, 4 are the access structure, the strong correlations are specified by the

nullifiers

ND − 2N1 + N3 + N4 = pD − (2p1 + q1) + (p3 − 2q3) + (p4 − 2q4)

and N1 − N3 − N4 = qD + p1 − (p3 − q3)− (p4 − q4) . (A.66)

The quantum correlations can be transferred to mode 1 by the 1,3,4 Decoding Sequence:(i) apply exp(−i(q1 + p3 − q3)) and exp(−i(q1 + p4 − q4)); (ii) followed by exp(ip1p3) and

exp(ip1p4); (iii) then exp(i2p21); (iv) finally F (π).

The covariance matrix of the state ρDA between mode D and mode 1 is given by

VDA;1,3,4 =

12σ2

D0 0 1

2σ2D

05+σ2σ2

D

2σ25

2σ2 0

0 52σ2

5+6σ4

2σ2 −2σ2

12σ2

D0 −2σ2 1+3σ2σ2

D

2σ2D

. (A.67)

After the parameter-estimation stage, local-squeezing is applied as in Eq. (A.51) ex-

cept mode h is now mode 1. The squeezing parameters in the current collaboration

are

γD =

√

σDσ

√

5 + σ2σ2D ; γ1 =

√

σ

5σD

√

5 + σ2σ2D . (A.68)

The covariance matrix of the unmeasured states then becomes

V ′DA;1,3,4 =

V(3,5) 0 0 c

0 V(3,5) c 0

0 c V ′q 0

c 0 0 V ′p

(A.69)

where c =√5/2σσD, and the variances of mode 1 are given by

V ′q =

(5 + 6σ4)√

5 + σ2σ2D

10σσD; V ′

p =5(1 + 3σ2σ2

D)

2σσD√

5 + σ2σ2D

. (A.70)

state-averaging ensues to balance the correlations of pD − q1 and qD −p1. The covari-

167

Chapter A. Appendix

Figure A.4: Secret sharing rate of CQ QSS protocols using CV cluster states withdifferent squeezing parameters σ. The squeezing parameter of dealer’s mode is set as σD =σ. Left panel: (2,3)-protocol for 2,3 collaboration (solid line) and 1,3 collaboration(dashed line). Right panel: (3,5)-protocol for 1,2,3 collaboration (solid line) and 1,3,4collaboration (dashed line).

ance matrix becomes

V ′′DA;1,3,4 =

V(3,5) 0 c 0

0 V(3,5) 0 −cc 0 VA;1,3,4 0

0 −c 0 VA;1,3,4

, (A.71)

where VA;1,3,4 = (V ′q + V ′

p)/2 with the definition in Eq. (A.70). Similar to the 1, 2, 3collaboration, the local coherent terms are eliminated by state-averaging.

After local q and p measurements, the results are treated as coming from a Gaussian

state. The variance of the dealer’s mode is recognized as V = V(3,5), and the analogous

channel parameters can be deduced as τ = c2/(V 2(3,5)−1/4) = 1 and χ = VA;1,3,4−V(3,5).

The minimal secret sharing rate can then be calculated by Eqs. (3.31)-(3.34) and (3.36).

The secret sharing rates of the (2,3)- and (3,5)-CQ protocol are plotted in Fig. A.4

for different σ.

168

Chapter A. Appendix

A.4 Example of QQ QSS

A.4.1 Example 1: (2,3)-QQ protocol

In the (2,3)-QQ protocol, the collaboration of any two out of the three parties can recover

the shared secret state with high fidelity, while any one party alone achieves much less

information about the secret. This protocol can be implemented by the same diamond-

shaped CV cluster state as that for the (2,3)-CQ protocol in Sec. A.3.1. In the infinitely

squeezed case, the nullifiers are given by Eq. (A.46), and these nullifiers with Eq. (3.7)

characterise the finitely squeezed cluster state.

Three different collaborations can be formed: parties 1 and 2, parties 1 and 3, and

parties 2 and 3. The state of the 1, 2 collaboration is local-unitarily equivalent to that

of the 1, 3 collaboration. As a result, the entanglement extracted between the dealer

and parties 1, 2 is the same as that between the dealer and parties 1, 3.For the 1, 2 collaboration, the quantum correlation can be transferred to mode 2 by

the 1,2 Decoding Sequence in Sec A.3.1.1. In the infinitely squeezing case, the nullifiers

in Eq. (A.47), which specifies the strong correlation, is transformed to pD−q2 and qD−p2.An infinitely squeezed two-mode cluster is hence extracted for teleportation. In the

finitely squeezing case, the 1,2 Decoding Sequence also transfers the strong correlation

to party 2. The covariance matrix of the extracted state between mode D and mode 2

is given by VDA;1,2 in Eq. (A.48).

For the 2, 3 collaboration, the quantum correlation, which is specified by the nul-

lifiers in Eq. (A.56), can be transferred to party 2 by applying a 50:50 beam splitter

between mode 2 and mode 3. An infinitely squeezed two-mode cluster state between

mode D and mode 2 is extracted in the infinitely squeezing case. While in the finitely

squeezing case, a strongly entangled state is extracted, of which the covariance matrix

VDA;2,3 is given in Eq. (A.48).

A.4.2 Example 2: (3,5)-QQ protocol

In the (3,5)-QQ protocol, any collaboration with three out of five parties can recover the

shared secret state with high fidelity, while fewer than three parties achieve much less

information about the secret. This protocol can be implemented by the same pentagonal

CV cluster state as used for the (3,5)-CQ protocol in Sec. A.3.2. In the infinitely squeezed

case, the nullifiers are given by Eq. (A.59), and these nullifiers with Eq. (3.7) characterise

the finitely squeezed cluster state.

Two categories of access structure can be formed: three neighbouring parties, and

169

Chapter A. Appendix

Figure A.5: Logarithmic negativity of the state extracted from a CV cluster state in QQQSS. Left panel: (2,3)-protocol for 2,3 collaboration (solid line) and 1,3 collaboration(dashed line). Right panel: (3,5)-protocol for 1,2,3 collaboration (solid line) and 1,3,4collaboration (dashed line).

two neighbours with one disjoint party. Within each category, the procedure of decoding

and the final entanglement extracted are the same for each collaboration.

Let us take the parties 1, 2, 3 as an example of the three neighbouring parties collab-

oration. The quantum correlation can be transferred to party 2 by the 1,2,3 Decoding

Sequence in Sec. A.3.2.1. In the infinitely squeezed case, the nullifiers in Eq. (A.60) that

specify the strong correlation are transformed to pD − q2 and qD − p2. This indicates

that an infinitely squeezed two-mode cluster is extracted in mode D and mode 2. While

in the finitely squeezed case, the decoding operation extracts a strongly entangled state

with the covariance matrix VDA;1,2,3 in Eq. (A.61).

On the other hand, parties 1, 3, 4 is an example of the two neighbours with one

disjoint party collaboration. The quantum correlations can be transferred to party 1

by the 1,3,4 Decoding Sequence in Sec A.3.2.2. In the infinitely squeezed case, an

infinitely squeezed two-mode cluster is extracted in mode D and mode 1, because the

nullifiers in Eq. (A.66) that specify the strong correlation is transformed to pD − q1 and

qD − p1. While in the finitely squeezing case, a strongly entangled state is extracted, and

its covariance matrix is given by VDA;1,3,4 in Eq. (A.67).

The logarithmic negativity of the extracted state for different collaborations in the

(2,3)- and the (3,5)-protocols is calculated by using Eq. (3.57) with the corresponding

covariance matrices. The result is plotted in Fig. A.5 against different squeezing param-

eters.

170

Chapter A. Appendix

A.5 Example of Application: Demonstration of Hong-

Ou-Mandel Effect

With the beam splitter and readout processes described in Ch. 6, numerous bosonic

effects can be simulated by the ion trap UBS. One of the particular important bosonic

phenomena is the HOM effect (see, e.g., ref. [14] for details), which shows the quantum

nature of bosons and becomes the foundation of some proposals of linear optics quantum

computation [101]. The HOM effect happens when two identical photons in different

input modes hit a 50:50 beam splitter at the same time. The resultant state will be a

superposition of two-photon state in either one of the output mode, i.e.

|1, 1〉 = a†1a†2|vac〉

50:50 beam splitter−−−−−−−−−−−→(

1

2a†21 − 1

2a†22

)

|vac〉 = 1√2|2, 0〉 − 1√

2|0, 2〉 . (A.72)

On the contrary to classical prediction that the photons will randomly distribute

to the output modes with equal probability, the above quantum state predicts that no

coincidence of photon can be measured in the output modes. The reason of this effect is

originated from the interference of photon state, which is related to the quantum nature

of bosons.

The HOM effect was first demonstrated experimentally in 1987 [86] by using photons

generated from spontaneous downward conversion. Various subsequent experiments have

been conducted by using photons from synchronized but less dependent sources [29], but

HOM effect is not demonstrated on systems other than optics. Thus, it is of great interest

to realise the HOM effect in other bosonic system, in order to verify if the HOM effect

is a generic bosonic behaviour, or it is simply an optical phenomenon that is not fully

understood. Furthermore, observing HOM effect in trapped ions is a strong evidence

of the motional states’ bosonic nature, which is still a theoretical prediction. In this

section, I outline the steps required to demonstrate HOM effect on a trapped-ion bosonic

simulator.

I consider a simulator that involves only two ions. Two phonon modes are available

for simulation: the centre-of-mass (CM) mode and the stretching mode. The circuit

diagram of the demonstration is shown in Fig. A.6. The ions are initially cooled to (or

very close to) the ground state. The simulation starts by initializing one phonon in each

mode by the following procedure: a blue sideband pulse is first applied to transfer the

state from |gg〉|0, 0〉 to |ge〉|1, 0〉, and then a carrier transition restores the electronic state

to |g〉, so the final state becomes |gg〉|1, 0〉 [127]. By repeating the above procedure for

the other mode, the total motional state becomes |gg〉|1, 1〉.

171

Chapter A. Appendix

B

B

BS BS

Preparation

Phonon

splitting

g,e

g,e

g,e

g,e

g

R

R

g,e

e,r

R

R

R

R

g,e

g,e

g,e

e,r

g,e,rg,e,r

Measurement

Figure A.6: Circuit diagram of the Hong-Ou-Mandel effect demonstration. Both thecentre-of-mass mode (left dashed line) and the stretching mode (right dashed line) areinvolved. The state is first prepared as |gg〉|1, 1〉 after the preparation stage, and thentransforms to |gg〉 1√

2(|0, 2〉 − |2, 0〉) by the beam splitter. The motional state of each

mode is transferred to the internal state of different ions, so both of the modes can bemeasured. If HOM effect is realized, final motional excitation will not be observed inboth the CM and stretching modes.

172

Chapter A. Appendix

Beam splitter operation is conducted by switching on the Hamiltonian (6.23) for

t = π/(2η1η2Ω), which changes the state to |ψ〉 = |gg〉(|2, 0〉 − |0, 2〉)/√2 2. The

next step is to measure the coincidence of phonon detection in the modes. If adia-

batic passage is employed, Raman field is applied to the first ion which the frequency

is tuned around the red sideband of the CM mode. The total motional state becomes

(|eg〉|1, 0〉− |gg〉|0, 2〉)/√2. Similarly, the Raman field of the second ion is tuned around

the red sideband of the stretching mode, and the state becomes (|eg〉|1, 0〉−|ge〉|0, 1〉)/√2.

Internal states of the ions are then measured by fluorescence measurement. If HOM ef-

fect is realised, one and only one of the ions is in the state |e〉. On the other hand,

if the final state contains a component of |gg〉|1, 1〉, it will be transformed to |ee〉|0, 0〉,which both ions are in the |e〉. Because the adiabatic transfer measurement method is

equivalent to a phonon number non-resolving detection, the coincidence of the phonon is

thus characterized by the rate of getting both ions in |e〉.Apart from using adiabatic transfer, phonon coincidence can also be measured by

the resonant pulses method. If the total phonon number in the system remains 2, and

if the beam splitter operation does not cause additional motional excitation, phonon

coincidence is solely contributed by the component |1, 1〉. The post-selection sequence

for measuring |1, 1〉 is as follow. Firstly, a red sideband pulse of CM mode is applied to

transfer |g 1〉 to |e 0〉. Then |g〉 is measured by fluorescence measurement. The exper-

iment is terminated when positive outcome is obtained (scattered photon is detected).

Otherwise, the remaining state is |e〉 associated with some phonon state. Particularly,

|e 0〉 comes from the original |g 1〉, which is the component related to the HOM effect.

A blue sideband pulse is then applied to transfer the |e 1〉 to |g 0〉 while leaving the

|e 0〉 unchanged. |g〉 is then measured again. If the system is assumed to involve only

2 phonons, then the measurement scheme can stop at this point, otherwise subsequent

blue sideband pulses are applied to clear the higher number Fock states associated to |e〉.The above procedure is repeated for the stretching mode with another ion. If the bosonic

simulator could realise the HOM effect, the probability of getting all negative results (no

photons detected in all measurement), which denotes the presence of |1, 1〉, would be very

small. In other words, experiments will be terminated with high probability due to the

measurement of |g〉.If there is one more metastable state available in each ion and if there are only two

phonons involved in the system, then post-selection is not required for the readout. The

operation described in section 6.5.2.2 can be conducted on both ions, which each ion

2Before the simulation, the action of beam splitter should be calibrated, for example by consideringits action on some single mode states.

173

Chapter A. Appendix

is responsible for one phonon mode. The operation transforms |gg〉(|02〉 − |20〉)/√2 to

(|gr〉 − |rg〉)/√2|0, 0〉, while the coincident state |gg〉|1, 1〉 will become |ee〉|0, 0〉. The

electronic states |g〉, |e〉, |r〉 of each ion are then measured. This process is equivalent to

measuring the PVM |0〉〈0|, |1〉〈1|, |2〉〈2| on the motional state. The occupation of the

phonon states can then be deduced by the statistics of the measurement results. Phonon

coincidence will be recognised if both ions are measured as |e〉 or |r〉. In other words, if

HOM effect exists, either one of the ion must be revealed to be in |g〉.

A.6 Moving multiple ions by harmonic trap

If N ions are trapped in a single harmonic potential with the trapping strength Ω2(t)

and centre R(t), the Hamiltonian of the system is

H =N∑

i

(

P 2i

2mi

+1

2Ω2(t)(xi − R(t))2

)

+N∑

i>j

e2

4πǫ0(xi − xj), (A.73)

where an ion with a larger index should have a larger displacement, i.e. 〈xi〉 > 〈xj〉 fori > j. Let the total state of the ions be |Ψ〉. The classical and quantum contribution of

the state can be decoupled by defining

|Ψ〉 ≡∏

i

Di(xi, pi)|ψ〉 , (A.74)

where Di is the displacement operator of the ith ion; |ψ〉 contains only the quantum

fluctuation; xi and pi are the classical position and momentum of the ith ions following

the classical equation of motion

xi =pimi

; pi = −Ω2(t)(xi−R)+∑

i>j

e2

4πǫ0r2ij−∑

i<j

e2

4πǫ0r2ijwhere rij = xi−xj . (A.75)

We note that our operation is equivalent to separating the classical and quantum contri-

butions by the expansion xi = xi + qi ; Pi = pi + pi, where qi and pi are the position and

momentum operators of the quantum fluctuation only.

The evolution of the quantum fluctuation follows

i∂t|ψ〉 = HQ|ψ〉 where (A.76)

HQ ≈∑

i

(

p2i2mi

+1

2Ω2(t)q2i

)

+∑

i>j

e2

4πǫ0r3ij(t)(qi − qj)

2 . (A.77)

174

Chapter A. Appendix

Here we have made an expansion of the Coulomb potential and collect only the second

order terms of position operators, i.e. the third or higher order terms of qi − qj are

neglected. This approximation is valid if the quantum fluctuation of each ion is much

narrower than the distance between ions, i.e. |〈xi〉 − 〈xj〉| ≫ 〈∆x2i 〉 for any i, j. In the

Heisenberg picture, a quadratic Hamiltonian evolves qi and pi only to combinations of

q’s and p’s [117], i.e.,

qi → U †QqiUQ =

∑

j

Aij(t)qj+Bij(t)pj ; pi → U †QpiUQ =

∑

j

Cij(t)qj+Dij(t)pj , (A.78)

where UQ is the corresponding evolution operator of HQ. Aij(t), Bij(t), Cij(t), Dij(t) are

real functions of time determining the multi-mode squeezing of the ions’ motion that

corresponds to the parametric excitation.

Using the identities

i∂t(U†QqiUQ) = U †

Q[qi, HQ]UQ and i∂t(U†QqiUQ) = U †

Q[pi, HQ]UQ , (A.79)

we can get the equations for A,B,C,D :

miAij = Cij ; (A.80)

Cij =[

− Ω2 − 2e2

4πǫ0

(

i∑

k=1

1

r3ik+

N∑

k=i

1

r3ki

)]

Aij +2e2

4πǫ0

(

i∑

k=1

1

r3ikAkj +

N∑

k=i

1

r3kiAkj

)

;

miBij = Dij ;

Dij =[

− Ω2 − 2e2

4πǫ0

(

i∑

k=1

1

r3ik+

N∑

k=i

1

r3ki

)]

Bij +2e2

4πǫ0

(

i∑

k=1

1

r3ikBkj +

N∑

k=i

1

r3kiBkj

)

.

Therefore under the quadratic approximation, the evolution operator, U , of H in

Eq. (A.73) transforms the state, the position and momentum of the ions as

|Ψ(t)〉 =(

∏

i

Di(xi, pi)

)

UQ|ψ(0)〉 . (A.81)

A.6.1 Motional Excitation

Assume the initial and final trapping frequency of the well are the same, i.e. Ω(0) =

Ω(T ) = Ω0, and the centre is moved from R(0) = 0 to R(T ). If the transportation is

ideal, the state of the ions, |Ψ(T )〉, is assumed to be the same as the state at t = 0 except

all the ions’ position are shifted by R(T ). In other words if we start from a ground

175

Chapter A. Appendix

state, we expect the final quantum fluctuation is in the ground state, the final classical

momentum is zero, and the final classical position of the ions are xi(0)+R(T ), satisfying

Ω20xi(0) =

∑

i>j

e2

4πǫ0r2ij(0)

−∑

i<j

e2

4πǫ0r2ij(0)

, where rij(0) = xi(0)− xj(0) . (A.82)

The motional excitation is defined as the excitation of quantum fluctuation with

respect to the ideal Hamiltonian

∑

i

(

p2i2mi

+1

2Ω2

0q2i

)

+∑

i>j

e2

4πǫ0r3ij(0)

(qi−qj)2 ≡∑

µ

( p2µ2mµ

+1

2Ω2

µq2µ

)

≡∑

µ

Ωµ√mµ

(

a†µaµ+1

2

)

,

(A.83)

which is the quadratic approximated HQ in Eq. (A.77) at t = 0. This quadratic Hamil-

tonian can be diagonalised with respect to the eigenmodes. We hereafter denote ions

with English alphabet indices (i, j, k, . . .) and eigenmodes with Greek alphabet indices

(µ, ν, . . .). Ωµ and mµ are the effective trapping frequency and the effective mass of the

mode µ. The position and momentum operator of the modes and ions are related by

linear combinations

qµ =∑

i

Eµiqi ; pµ =∑

i

Fµipi , (A.84)

where Eµi and Fµi are real constants determined by the eigenvectors of the diagonaliza-

tion.

The final excitation of the quantum fluctuation, represented by the state |ψ(T )〉,is defined by subtracting the expected classical displacement from the ions’ final state,

|Ψ(T )〉, i.e.,

|ψ(T )〉 ≡∏

i

(

D†i

(

xi(0) +R(T ), 0)

)

|Ψ(T )〉

=∏

i

(

D†i

(

xi(0) +R(T ), 0)

Di

(

xi(T ), pi(T ))

)

UQ|ψ(0)〉 , (A.85)

where the last identity comes from Eq. (A.81). Therefore the final excitation of the mode

µ is given by

nµ ≡ 〈ψ(T )|a†µaµ|ψ(T )〉 (A.86)

= 〈ψ(0)|U †Q

(

∏

i

D†i (∆xi, pi)

)( p2µ2√mµΩµ

+1

2

√mµΩµq

2µ −

1

2

)(

∏

i

Di(∆xi, pi))

UQ|ψ(0)〉 ,

where ∆xi ≡ xi(T )− xi(0)− R(T ) is the position displacement mismatch.

176

Chapter A. Appendix

We tackle the terms in Eq. (A.87) one by one. The displacement operator Di trans-

forms the ions’ position and momentum operators as

D†i (∆xi, pi)qiDi(∆xi, pi) = qi +∆xi ; D†

i (∆xi, pi)piDi(∆xi, pi) = pi + pi , (A.87)

and the modes’ operators as

(

∏

i

D†i (∆xi, pi)

)

qµ

(

∏

i

Di(∆xi, pi))

=∑

i

Eµiqi +∑

i

Eµi∆xi ≡ qµ + xµ

(

∏

i

D†i (∆xi, pi)

)

pµ

(

∏

i

Di(∆xi, pi))

=∑

i

Eµipi +∑

i

Eµipi ≡ pµ + pµ . (A.88)

Consider the initial ground state satisfies 〈qµ〉0 = 〈pµ〉0 = 0, where we define for any

operator O, 〈O〉 ≡ 〈ψ(0)|O|ψ(0)〉. The motional excitation can be separated as nµ =

ncoherentµ + nparametric

µ , where

ncoherentµ =

p2µ2√mµΩµ

+1

2

√mµΩµx

2µ ; (A.89)

nparametricµ = 〈ψ(0)|U †

Q

( p2µ2√mµΩµ

+1

2

√mµΩµq

2µ

)

UQ|ψ(0)〉 −1

2. (A.90)

The first term is the coherent excitation, and the second term is the parametric excitation.

Combining Eq. (A.81) and (A.84), the mode operators transform under UQ as

U †QqµUQ =

∑

ijν

EµiAijE−1jν qν + EµiBijF

−1jν pν ≡

∑

ν

A′µν qν +B′

µν pν

U †QpµUQ =

∑

ijν

FµiCijE−1jν qν + FµiDijF

−1jν pν ≡

∑

ν

C ′µν qν +D′

µν pν , (A.91)

where E−1 and F−1 are the inverse of E and F respectively. Putting into Eq. (A.89),

the parametric excitation is given by

nparametricµ =

∑

ν

(C ′µν)

2〈q2ν〉0 + (D′µν)

2〈p2ν〉02√mµΩµ

+1

2

√mµΩµ

(

(A′µν)

2〈q2ν〉0 + (B′µν)

2〈p2ν〉0)

− 1

2

(A.92)

where we have used the fact that 〈qµpν + pµqν〉0 = 0 for any µ and ν. We note that

although the initial state is assumed to be the ground state, but our calculation can be

easily extended to general mixed initial state.

Our method has an advantage that the evolution of N ions can be obtained by

numerically solving 2N2 differential equation of real functions. In conventional method,

177

Chapter A. Appendix

the final excitation of the transportation has to be calculated by solving the coupled

evolution of the states of N phononic modes, which the Hilbert space will be large if

the modes can be significantly excited during the diabatic transportation. We note that

the effect of anharmonic terms can be calculated perturbatively using the squeezing

parameters we have calculated [18].

A.6.2 Example: Two ions in a trap

If two different ions trapping in a single well is transported, the evolution is governed by

the Hamiltonian

H =P 21

2m1

+P 22

2m2

+1

2Ω2(t)(x1 −R)2 +

1

2Ω2(t)(x2 −R)2 +

e2

4πǫ0(x2 − x1). (A.93)

The classical position and momentum can be extracted, and they satisfy the classical

equation of motion

x1 =p1m1

; p1 = −Ω2(t)(

x1 − R(t))

− e2

4πǫ0(x2 − x1)2;

x2 =p2m2

; p2 = −Ω2(t)(

x2 − R(t))

+e2

4πǫ0(x2 − x1)2. (A.94)

The quantum contribution satisfies Eq. (A.76), where the quadratic approximated HQ is

HQ ≈ p212m1

+p222m2

+1

2Ω2(t)q21 +

1

2Ω2(t)q22 +

e2

4πǫ0(x2 − x1)3(q1 − q2)

2 . (A.95)

In the Heisenberg picture, the evolution operator of HQ transforms the position and

momentum operator as

U †Qq1UQ = A11q1 + A12q2 +B11p1 +B12p2 ;

U †Qq2UQ = A21q1 + A22q2 +B21p1 +B22p2 ;

U †Qp1UQ = C11q1 + C12q2 +D11p1 +D12p2 ;

U †Qp2UQ = C21q1 + C22q2 +D21p1 +D22p2 , (A.96)

178

Chapter A. Appendix

where the variation of the squeezing parameters satisfy

miAij = Cij ; miBij = Dij

C11 =(

− Ω2(t)− 2e2

4πǫ0r3(t)

)

A11 +2e2

4πǫ0r3(t)A21 ;

D11 =(

− Ω2(t)− 2e2

4πǫ0r3(t)

)

B11 +2e2

4πǫ0r3(t)B21 ;

C21 =(

− Ω2(t)− 2e2

4πǫ0r3(t)

)

A21 +2e2

4πǫ0r3(t)A11 ;

D21 =(

− Ω2(t)− 2e2

4πǫ0r3(t)

)

B21 +2e2

4πǫ0r3(t)B11 ;

C12 =(

− Ω2(t)− 2e2

4πǫ0r3(t)

)

A12 +2e2

4πǫ0r3(t)A22 ;

D12 =(

− Ω2(t)− 2e2

4πǫ0r3(t)

)

B12 +2e2

4πǫ0r3(t)B22 ;

C22 =(

− Ω2(t)− 2e2

4πǫ0r3(t)

)

A22 +2e2

4πǫ0r3(t)A12 ;

D22 =(

− Ω2(t)− 2e2

4πǫ0r3(t)

)

B22 +2e2

4πǫ0r3(t)B12 . (A.97)

At t = 0, the classical separation between the ions is

Ω2(0)x1(0) = − e2

4πǫ0(x2(0)− x1(0))2; Ω2(0)x2(0) =

e2

4πǫ0(x2(0)− x1(0))2. (A.98)

The vibrational modes α and β can be defined by diagonalising the HQ in Eq. (A.95) at

t = 0, viz. p2α2mα

+p2β2mβ

+ 12Ω2

αq2α + 1

2Ω2

β q2β , where the mode operators are defined as:

qα = Eα1q1 + Eα2q2 ; qβ = Eβ1q1 + Eβ2q2 ; pα = Fα1p1 + Fα2p2 ; pβ = Fβ1p1 + Eβ2p2 ,

(A.99)

179

Chapter A. Appendix

where

Eα1 =m1 −m2 +

√

m21 −m1m2 +m2

2

m2; Eα2 = 1 ;

Eβ1 =m1 −m2 −

√

m21 −m1m2 +m2

2

m2; Eβ2 = 1 ;

mα = 1/(E2

α1

m1+E2

α2

m2

)

; mβ = 1/(E2

β1

m1+E2

β2

m2

)

;

Ω2α =

mα(m1 +m2 −√

m21 −m1m2 +m2

2)

m1m2Ω2

0 ;

Ω2β =

mβ(m1 +m2 +√

m21 −m1m2 +m2

2)

m1m2Ω2

0 ;

Fα1 =mα

m1Eα1 ; Fα2 =

mα

m2Eα2 ;

Fβ1 =mβ

m1

Eβ1 ; Fβ2 =mβ

m2

Eβ2 .

We note that both ions move in phase for mode α, while they move out of phase for

mode β.

Using the ions’ final classical position and momentum at t = T , obtained by solving

Eq. (A.94), and the relations to the modes’ classical position and momentum in Eq. (A.88)

and (A.99), the coherent excitation of any mode can be calculated by Eq. (A.89). The

parametric excitation of any mode can be calculated by Eq. (A.91), (A.92), (A.99) and

the squeezing parameters A,B,C,D at t = T , obtained by solving Eq. (A.97) with the

initial conditions

A11(0) = A22(0) = D11(0) = D22(0) = 1 ; others = 0 at t = 0 . (A.100)

180

Bibliography

[1] see e.g., http://www.idquantique.com/, http://www.magiqtech.com/

MagiQ/Home.html and http://www.scientificamerican.com/article/

swiss-test-quantum-cryptography/.

[2] D. Gottesman, 1997, Ph.D. thesis (California Institute of Technology).

[3] R. Garcıa-Patron, 2007, Ph.D. thesis (Universite Libre de Bruxelles).

[4] T. C. Ralph, and A. P. Lund, 2009, in Proceedings of the 9th International Con-

ference on Quantum Communication Measurement and Computing, edited by A.

Lvovsky (AIP, New York), p. 155160.

[5] Stephen M. Barnett, Quantum Information (Oxford University Press, New York,

U.S.A., 2009).

[6] Richard L. Liboff, Introductory Quantum Mechanics, (Addison Wesley, San Fran-

cisco, U.S.A., 2003).

[7] Naum Ilich Akhiezer, The calculus of variations, (Blaisdell Publishing Company,

New York, U.S.A., 1962).

[8] NIST atomic spectra database, http://www.nist.gov/pml/data/asd.cfm.

[9] B. H. Bransden and C. J. Joachain, Physics of atoms and molecules (Pearson

Education, 2003).

[10] L. Allen and J. H. Eberly, Optical Resonance and Two- Level Atoms, (Dover, New

York, 1987).

[11] see e.g. http://qwiki.stanford.edu/index.php/Complexity_Zoo.

[12] Simple discussions of energy transfer in coupled classical oscillators can be

found in, e.g. http://physics-animations.com/Physics/English/link_txt.

181

http://www.idquantique.com/

http://www.magiqtech.com/MagiQ/Home.html

http://www.magiqtech.com/MagiQ/Home.html

http://www.scientificamerican.com/article/swiss-test-quantum-cryptography/

http://www.scientificamerican.com/article/swiss-test-quantum-cryptography/

http://www.nist.gov/pml/data/asd.cfm

http://qwiki.stanford.edu/index.php/Complexity_Zoo

http://physics-animations.com/Physics/English/link_txt.htm

BIBLIOGRAPHY BIBLIOGRAPHY

htm and http://www.theorphys.science.ru.nl/people/fasolino/sub_java/

pendula/doublependul-en.shtml.

[13] A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw-

Hill Inc., USA, 1965).

[14] C. Gerry and P. Knight, Introductory Quantum Optics, (Cambridge University

Press, New York, 2005).

[15] D.M. Greenberger, M. Horne, and A. Zeilinger, in Bell’s Theorem, Quantum The-

ory, and Conceptions of the Universe, edited by M. Kafatos (Kluwer, Dordrecht,

1989).

[16] S. Aaronson. Quantum copy-protection and quantum money. In Computational

Complexity, 2009. CCC ’09. 24th Annual IEEE Conference on, pages 229–242,

July 2009.

[17] Scott Aaronson and Alex Arkhipov. The computational complexity of linear optics.

Theory of Computing, 9(4):143–252, 2013.

[18] Sumiyoshi Abe and Rudolf Ehrhardt. Effects of anharmonicity on nonclassical

states of the time-dependent harmonic oscillator. Physical Review A, 48(2):986,

1993.

[19] Joseba Alonso, Florian Leupold, Ben Keitch, and Jonathan Home. Quantum con-

trol of the motional states of trapped ions through fast switching of trapping po-

tentials. New Journal of Physics, 15(2):023001, 2013.

[20] G Van Assche, J Cardinal, and N J Cerf. Reconciliation of a quantum-distributed

Gaussian key. IEEE Transactions on Information Theory, 50(2):394–400, 2004.

[21] M D Barrett, J Chiaverini, T Schaetz, J Britton, W M Itano, J D Jost, E Knill,

C Langer, D Leibfried, R Ozeri, and D J Wineland. Deterministic quantum tele-

portation of atomic qubits. Nature, 429(6993):737–739, 2004.

[22] M D Barrett, B DeMarco, T Schaetz, V Meyer, D Leibfried, J Britton, J Chiaverini,

W M Itano, B Jelenkovic, J D Jost, C Langer, T Rosenband, and D J Wineland.

Sympathetic cooling of 9Be+ and 24Mg+ for quantum logic. Physical Review A,

68(4):042302, Jan 2003.

182

http://physics-animations.com/Physics/English/link_txt.htm

http://www.theorphys.science.ru.nl/people/fasolino/sub_java/pendula/doublependul-en.shtml

http://www.theorphys.science.ru.nl/people/fasolino/sub_java/pendula/doublependul-en.shtml


[23] S Beigi and R Konig. Simplified instantaneous non-local quantum computation with

applications to position-based cryptography. New Journal of Physics, 13(9):093036,

2011. 18 pages, 2 figures.

[24] John S. Bell. On the Einstein Podolsky Rosen paradox. Physics, 1(3):195–200, Feb

1964.

[25] John S. Bell. On the problem of hidden variables in quantum mechanics. Reviews

of Modern Physics, 38(3):447–452, 1966.

[26] C H Bennett and G Brassard. Quantum cryptography: Public key distribution

and coin tossing. In Proceedings of IEEE International Conference on Computers,

Systems and Signal Processing, pages 175–179, 1984.

[27] C H Bennett, G Brassard, C Crepeau, R Jozsa, A Peres, and W K Wootters.

Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-

Rosen channels. Physical Review Letters, 70(13):1895–1899, Jan 1993.

[28] Charles H Bennett, David P Divincenzo, Peter W Shor, John A Smolin, Barbara M

Terhal, and William K Wootters. Remote state preparation. Physical Review

Letters, 87(7):077902, Jul 2001.

[29] J Beugnon, M P A Jones, J Dingjan, B Darquie, G Messin, A Browaeys, and

P Grangier. Quantum interference between two single photons emitted by inde-

pendently trapped atoms. Nature, 440(7085):779–782, 2006.

[30] R B Blakestad, C Ospelkaus, A P VanDevender, J M Amini, J Britton, D Leibfried,

and D. J Wineland. High-fidelity transport of trapped-ion qubits through an x-

junction trap array. Physical Review Letters, 102(15):153002, Jan 2009.

[31] G R Blakley. Safeguarding cryptographic keys. Proceedings of the National Com-

puter Conference, 48:313–317, Jan 1979.

[32] Rainer Blatt and David Wineland. Entangled states of trapped atomic ions. Nature,

453(7198):1008–1015, Jan 2008.

[33] R Bowler, J Gaebler, Y Lin, T R Tan, D Hanneke, J D Jost, J P Home, D Leibfried,

and D J Wineland. Coherent diabatic ion transport and separation in a multizone

trap array. Physical Review Letters, 109(8):080502, 2012.

[34] S L Braunstein and P van Loock. Quantum information with continuous variables.

Reviews of Modern Physics, 77(2):513–577, Jan 2005.

183


[35] K Brown, A Wilson, Y Colombe, C Ospelkaus, A Meier, E Knill, D Leibfried, and

D Wineland. Single-qubit-gate error below 104 in a trapped ion. Physical Review

A, 84(3):030303, Sep 2011.

[36] K R Brown, C Ospelkaus, Y Colombe, A C Wilson, D Leibfried, and D J Wineland.

Coupled quantized mechanical oscillators. Nature, 471(7337):196–199, Jan 2011.

[37] D E Browne and T Rudolph. Resource-efficient linear optical quantum computa-

tion. Physical Review Letters, 95(1):010501, Jan 2005.

[38] Daniel E Browne, Jens Eisert, Stefan Scheel, and Martin B Plenio. Driving non-

Gaussian to Gaussian states with linear optics. Physical Review A, 67:062320, Jun

2003.

[39] H. Buhrman, N. Chandran, S. Fehr, R. Gelles, V. Goyal, R. Ostrovsky, and

C. Schaffner. Position-based quantum cryptography: Impossibility and construc-

tions. SIAM Journal on Computing, 43(1):150–178, 2014.

[40] Harry Buhrman, Serge Fehr, Christian Schaffner, and Florian Speelman. The

garden-hose model. In Proceedings of the 4th Conference on Innovations in The-

oretical Computer Science, ITCS ’13, pages 145–158, New York, NY, USA, 2013.

ACM.

[41] I Buluta and F Nori. Quantum simulators. Science, 326(5949):108, 2009.

[42] N Chandran, V Goyal, R Moriarty, and R Ostrovsky. Position based cryptography.

Advances in Cryptology-CRYPTO 2009, pages 391–407, 2009.

[43] Nishanth Chandran, Serge Fehr, Ran Gelles, Vipul Goyal, and Rafail Ostrovsky.

Position-based quantum cryptography. arXiv:quant-ph/1005.1750, May 2010.

[44] Xi Chen, I Lizuain, A Ruschhaupt, D Guery-Odelin, and J Muga. Shortcut

to adiabatic passage in two- and three-level atoms. Physical Review Letters,

105(12):123003, Sep 2010.

[45] Xi Chen, A Ruschhaupt, S Schmidt, A del Campo, D Guery-Odelin, and J. G Muga.

Fast optimal frictionless atom cooling in harmonic traps: Shortcut to adiabaticity.

Physical Review Letters, 104(6):063002, Jan 2010.

[46] Xi Chen, E Torrontegui, Dionisis Stefanatos, Jr-Shin Li, and J G Muga. Optimal

trajectories for efficient atomic transport without final excitation. Physical Review

A, 84(4):043415, Jan 2011.

184


[47] Y A Chen, A N Zhang, Z Zhao, X Q Zhou, C Y Lu, C Z Peng, T Yang, and J W

Pan. Experimental quantum secret sharing and third-man quantum cryptography.

Physical Review Letters, 95(20):200502, 2005.

[48] Ran Hee Choi, Ben Fortescue, Gilad Gour, and Barry C Sanders. Entangle-

ment sharing protocol via quantum error-correcting codes. Physical Review A,

87(3):032319, Mar 2013.

[49] J I Cirac, R Blatt, and P Zoller. Nonclassical states of motion in a three-dimensional

ion trap by adiabatic passage. Physical Review A, 49(5):3174–3177, 1994.

[50] J I Cirac and P Zoller. Quantum computations with cold trapped ions. Physical

Review Letters, 74(20):4091–4094, 1995.

[51] R Cleve, D Gottesman, and H K Lo. How to share a quantum secret. Physical

Review Letters, 83(3):648–651, Jan 1999.

[52] D Dieks. Communication by EPR devices. Physics Letters A, 92(6):271–272, 1982.

[53] David P DiVincenzo. The physical implementation of quantum computation.

Fortschritte der Physik, 48(9-11):771–783, Oct 2000.

[54] L M Duan, G Giedke, J I Cirac, and P Zoller. Inseparability criterion for continuous

variable systems. Physical Review Letters, 84(12):2722–2725, Jan 2000.

[55] L M Duan and C Monroe. Colloquium: Quantum networks with trapped ions.

Reviews of Modern Physics, 82(2):1209–1224, Jan 2010.

[56] L. M. Duan and R Raussendorf. Efficient quantum computation with probabilistic

quantum gates. Physical Review Letters, 95(8):080503, 2005.

[57] Albert Einstein, Boris Podolsky, and Nathan Rosen. Can quantum-mechanical

description of physical reality be considered complete? Physical Review, 47(10):777,

1935.

[58] J Eisert, D E Browne, S Scheel, and M B Plenio. Distillation of continuous-variable

entanglement with optical means. Annals of Physics, 311(2):431–458, 2004.

[59] J Eisert, S Scheel, and M B Plenio. Distilling Gaussian states with Gaussian

operations is impossible. Physical Review Letters, 89(13):137903, Jan 2002.

[60] A K Ekert. Quantum cryptography based on Bell’s theorem. Physical Review

Letters, 67(6):661–663, 1991.

185


[61] R P Feynman. Simulating physics with computers. International Journal of The-

oretical Physics, 21(6):467–488, 1982.

[62] Jaromır Fiurasek. Gaussian transformations and distillation of entangled Gaussian

states. Physical Review Letters, 89(13):137904, Sep 2002.

[63] Jaromır Fiurasek, Ladislav Mista, and Radim Filip. Entanglement concentration

of continuous-variable quantum states. Physical Review A, 67(2):022304, 2003.

[64] Steven T Flammia, Nicolas C Menicucci, and Oliver Pfister. The optical frequency

comb as a one-way quantum computer. Journal of Physics B: Atomic, Molecular

and Optical Physics, 42(11):114009, 2009.

[65] A G Fowler, S J Devitt, and L C L Hollenberg. Implementation of Shor’s algorithm

on a linear nearest neighbour qubit array. Quantum Information and Computation,

4(4):237–251, Jan 2004.

[66] J R Fuhr and W L Wiese. Tables of atomic transition probabilities for beryllium

and boron. Journal of Physical and Chemical Reference Data, 39:013101, 2010.

[67] Omar Gamel and Daniel F V James. Time-averaged quantum dynamics and the

validity of the effective hamiltonian model. Physical Review A, 82(5):052106, Nov

2010.

[68] Raul Garcıa-Patron and Nicolas J. Cerf. Unconditional optimality of Gaussian

attacks against continuous-variable quantum key distribution. Physical Review

Letters, 97(19):190503, 2006.

[69] S A Gardiner, J I Cirac, and P Zoller. Nonclassical states and measurement of

general motional observables of a trapped ion. Physical Review A, 55(3):1683,

1997.

[70] R Gerritsma, G Kirchmair, F Zahringer, E Solano, R Blatt, and C F Roos. Quan-

tum simulation of the dirac equation. Nature, 463(7277):68–71, 2010.

[71] D Gottesman. Theory of quantum secret sharing. Physical Review A, 61(4):042311,

2000.

[72] M Greiner, O Mandel, T Esslinger, T. W Hansch, and I Bloch. Quantum phase

transition from a superfluid to a mott insulator in a gas of ultracold atoms. Nature,

415(6867):39–44, 2002.

186


[73] L K Grover. Quantum mechanics helps in searching for a needle in a haystack.

Physical Review Letters, 79(2):325–328, Jan 1997.

[74] Mile Gu, Christian Weedbrook, Nicolas C. Menicucci, Timothy C. Ralph, and Peter

van Loock. Quantum computing with continuous-variable clusters. Physical Review

A, 79(6):062318, Jun 2009.

[75] S Gulde, M Riebe, G P T Lancaster, C Becher, J Eschner, H Haffner, F Schmidt-

Kaler, I L Chuang, and R Blatt. Implementation of the Deutsch–Jozsa algorithm

on an ion-trap quantum computer. Nature, 421(6918):48–50, 2003.

[76] Robert H Hadfield. Single-photon detectors for optical quantum information ap-

plications. Nature Photonics, 3(12):696–705, Jan 2009.

[77] H Haffner, W Hansel, C F Roos, J Benhelm, D Chek al kar, M Chwalla, T Korber,

U D Rapol, M Riebe, P O Schmidt, C Becher, O Guhne, W Dur, and R Blatt.

Scalable multiparticle entanglement of trapped ions. Nature, 438(7068):643–646,

Jan 2005.

[78] H Haffner, C F Roos, and R Blatt. Quantum computing with trapped ions. Physics

Reports, Jan 2008.

[79] M Harlander, R Lechner, M Brownnutt, R Blatt, and W Haensel. Trapped-ion

antennae for the transmission of quantum information. Nature, 471(7337):200–203,

Jan 2011.

[80] D J Heinzen and D J Wineland. Quantum-limited cooling and detection of radio-

frequency oscillations by laser-cooled ions. Physical Review A, 42(5):2977, 1990.

[81] W K Hensinger, S Olmschenk, D Stick, D Hucul, M Yeo, M Acton, L Deslauriers,

C Monroe, and J Rabchuk. T-junction ion trap array for two-dimensional ion

shuttling, storage, and manipulation. Applied Physics Letters, 88(3):034101, 2006.

[82] W K Hensinger, D W Utami, H S Goan, K Schwab, C Monroe, and G J Milburn.

Ion trap transducers for quantum electromechanical oscillators. Physical Review

A, 72(4):041405, Jan 2005.

[83] M Hillery, V Buzek, and A Berthiaume. Quantum secret sharing. Physical Review

A, 59(3):1829–1834, Jan 1999.

[84] A S Holevo. Bounds for the quantity of information transmitted by a quantum

communication channel. Problemy Peredachi Informatsii, 9(3):3–11, 1973.

187


[85] J P Home and A M Steane. Electrode configurations for fast separation of trapped

ions. Quantum Information and Computation, 6(4-5):289–325, Jan 2006.

[86] C K Hong, Z Y Ou, and L Mandel. Measurement of subpicosecond time intervals

between two photons by interference. Physical Review Letters, 59(18):2044–2046,

1987.

[87] Zbigniew Idziaszek, Tommaso Calarco, and Peter Zoller. Ion-assisted ground-state

cooling of a trapped polar molecule. Physical Review A, 83(5):053413, May 2011.

[88] Satoshi Ishizaka and Tohya Hiroshima. Asymptotic teleportation scheme as a uni-

versal programmable quantum processor. Physical Review Letters, 101(24):240501,

Jan 2008.

[89] Satoshi Ishizaka and Tohya Hiroshima. Quantum teleportation scheme by selecting

one of multiple output ports. Physical Review A, 79(4):042306, Jan 2009.

[90] D. F. V. James. Quantum dynamics of cold trapped ions with application to

quantum computation. Applied Physics B: Lasers and Optics, 66(2):181–190, 1998.

[91] Daniel F. V. James. Quantum computation with hot and cold ions: An assessment

of proposed schemes. Fortschr Phys, 48(9-11):823–837, Jan 2000.

[92] M Johanning, A. F Varon, and C Wunderlich. Quantum simulations with cold

trapped ions. Journal of Physics B: Atomic, Molecular and Optical Physics,

42:154009, 2009.

[93] J D Jost, J P Home, J M Amini, D Hanneke, R Ozeri, C Langer, J J Bollinger,

D Leibfried, and D J Wineland. Entangled mechanical oscillators. Nature,

459(7247):683–U84, Jan 2009.

[94] M J Kastoryano, F Reiter, and A S Sørensen. Dissipative preparation of entangle-

ment in optical cavities. Physical Review Letters, 106(9):090502, Feb 2011.

[95] Adrian Keet, Ben Fortescue, Damian Markham, and Barry C Sanders. Quantum

secret sharing with qudit graph states. Physical Review A, 82(6):062315, Jan 2010.

[96] A P Kent, W J Munro, T P Spiller, and R G Beausoleil. Tagging systems, 2006.

US patent US20067075438.

188


[97] Adrian Kent, William J Munro, and Timothy P Spiller. Quantum tagging: Au-

thenticating location via quantum information and relativistic signaling constraints.

Physical Review A, 84(1):012326, Jan 2011.

[98] D Kielpinski, B E King, C J Myatt, C A Sackett, Q A Turchette, W M Itano,

C Monroe, D J Wineland, and W H Zurek. Sympathetic cooling of trapped ions

for quantum logic. Physical Review A, 61(3):032310, 2000.

[99] D Kielpinski, C Monroe, and D J Wineland. Architecture for a large-scale ion-trap

quantum computer. Nature, 417(6890):709–711, 2002.

[100] H C Kim, M H Lee, J Y Ji, and J K Kim. Heisenberg-picture approach to the exact

quantum motion of a time-dependent forced harmonic oscillator. Physical Review

A, 53(6):3767–3772, Jan 1996.

[101] E Knill, R Laflamme, and G J Milburn. A scheme for efficient quantum computa-

tion with linear optics. Nature, 409(6816):46–52, 2001.

[102] A M Lance, T Symul, W P Bowen, B C Sanders, and P K Lam. Tripartite quantum

state sharing. Physical Review Letters, 92(17):177903, Jan 2004.

[103] C Langer, R Ozeri, J D Jost, J Chiaverini, B DeMarco, A Ben-Kish, R B Blakestad,

J Britton, D B Hume, and W M Itano. Long-lived qubit memory using atomic ions.

Physical Review Letters, 95(6):060502, 2005.

[104] B P Lanyon, C Hempel, D Nigg, M Muller, R Gerritsma, F Zahringer, P Schindler,

J T Barreiro, M Rambach, G Kirchmair, M Hennrich, P Zoller, R Blatt, and

C F Roos. Universal digital quantum simulation with trapped ions. Science,

334(6052):57–61, Oct 2011.

[105] Hoi-Kwan Lau. Rapid laser-free ion cooling by controlled collision. arXiv:quant-

ph/1212.5628.

[106] Hoi-Kwan Lau and Daniel F V James. Decoherence and dephasing errors caused

by d.c. stark effect in rapid ion transport. Physical Review A, 83:062330, Jun 2011.

[107] Hoi-Kwan Lau and Daniel F V James. Proposal for a scalable universal bosonic

simulator using individually trapped ions. Physical Review A, 85(6):062329, 2012.

[108] Hoi-Kwan Lau and Hoi-Kwong Lo. Insecurity of position-based quantum-

cryptography protocols against entanglement attacks. Physical Review A,

83(1):012322, Jan 2011.

189


[109] Hoi-Kwan Lau and Christian Weedbrook. Quantum secret sharing with continuous-

variable cluster states. Physical Review A, 88(4):042313, 2013.

[110] D Leibfried, R Blatt, C Monroe, and D Wineland. Quantum dynamics of single

trapped ions. Reviews of Modern Physics, 75(1):281, 2003.

[111] D Leibfried, B DeMarco, V Meyer, D Lucas, M Barrett, J Britton, W M Itano,

B Jelenkovic, C Langer, T Rosenband, and D J Wineland. Experimental demon-

stration of a robust, high-fidelity geometric two ion-qubit phase gate. Nature,

422(6930):412–415, Jan 2003.

[112] D Leibfried, B DeMarco, V Meyer, M Rowe, A Ben-Kish, J Britton, W M Itano,

B Jelenkovic, C Langer, T Rosenband, and D J Wineland. Trapped-ion quantum

simulator: Experimental application to nonlinear interferometers. Physical Review

Letters, 89(24):247901, Jan 2002.

[113] H R Lewis and W B Riesenfeld. An exact quantum theory of the time dependent

harmonic oscillator and of a charged particle in a time dependent electromagnetic

field. Journal of Mathematical Physics, 10:1458, 1969.

[114] Daniel Lidar, Isaac Chuang, and K Whaley. Decoherence-free subspaces for quan-

tum computation. Physical Review Letters, 81(12):2594, 1998.

[115] Y Lin, J P Gaebler, T R Tan, R Bowler, J D Jost, D Leibfried, and D J Wineland.

Sympathetic electromagnetically-induced-transparency laser cooling of motional

modes in an ion chain. Physical Review Letters, 110(15):153002, Apr 2013.

[116] S Lloyd. Universal quantum simulators. Science, 273(5278):1073–1078, Jan 1996.

[117] S Lloyd and S L Braunstein. Quantum computation over continuous variables.

Physical Review Letters, 82(8):1784–1787, Jan 1999.

[118] C F Lo. Generating displaced and squeezed number states by a general driven

time-dependent oscillator. Physical Review A, 43(1):404–409, Jan 1991.

[119] H K Lo and H F Chau. Is quantum bit commitment really possible? Physical


[120] H K Lo and H F Chau. Unconditional security of quantum key distribution over

arbitrarily long distances. Science, 283(5410):2050–2056, Jan 1999.

190


[121] S Machnes, M B Plenio, B Reznik, A M Steane, and A Retzker. Superfast laser

cooling. Physical Review Letters, 104(18):183001, Jan 2010.

[122] R.A Malaney. Quantum location verification in noisy channels. In Global Telecom-

munications Conference (GLOBECOM 2010), 2010 IEEE, pages 1–6, Dec 2010.

[123] Robert A Malaney. Location-dependent communications using quantum entangle-

ment. Physical Review A, 81(4):042319, Jan 2010.

[124] Damian Markham and Barry C Sanders. Graph states for quantum secret sharing.

Physical Review A, 78(4):042309, Jan 2008.

[125] D Mayers. Unconditionally secure quantum bit commitment is impossible. Physical


[126] Dominic Mayers. Unconditional security in quantum cryptography. J. ACM,

48(3):351–406, May 2001.

[127] D M Meekhof, C Monroe, B E King, W M Itano, and D J Wineland. Genera-

tion of nonclassical motional states of a trapped atom. Physical Review Letters,

76(11):1796–1799, 1996.

[128] N C Menicucci, P van Loock, M Gu, C Weedbrook, T C Ralph, and M A Nielsen.

Universal quantum computation with continuous-variable cluster states. Physical

Review Letters, 97(11):110501, 2006.

[129] Nicolas C. Menicucci, Steven T. Flammia, and Olivier Pfister. One-way quantum

computing in the optical frequency comb. Physical Review Letters, 101(13):130501,

Sep 2008.

[130] Nicolas C Menicucci, Steven T Flammia, Hussain Zaidi, and Olivier Pfister. Ul-

tracompact generation of continuous-variable cluster states. Physical Review A,

76(1):010302, Jul 2007.

[131] Nicolas C Menicucci, Xian Ma, and Timothy C Ralph. Arbitrarily large continuous-

variable cluster states from a single quantum nondemolition gate. Physical Review

Letters, 104(25):250503, Jun 2010.

[132] David Menzies and Natalia Korolkova. Weak values and continuous-variable en-

tanglement concentration. Physical Review A, 76(6):062310, 2007.

191


[133] D L Moehring, P Maunz, S Olmschenk, K C Younge, D N Matsukevich, L M Duan,

and C Monroe. Entanglement of single-atom quantum bits at a distance. Nature,

449(7158):68–71, 2007.

[134] C Monroe, D M Meekhof, B E King, and D J Wineland. A “Schrodinger cat”

superposition state of an atom. Science, 272(5265):1131, 1996.

[135] C Monroe, R Raussendorf, A Ruthven, K. R Brown, P Maunz, L.-M Duan, and

J Kim. Large-scale modular quantum-computer architecture with atomic memory

and photonic interconnects. Physical Review A, 89(2):022317, Feb 2014.

[136] Thomas Monz, Philipp Schindler, Julio T Barreiro, Michael Chwalla, Daniel Nigg,

William A Coish, Maximilian Harlander, Wolfgang Haensel, Markus Hennrich, and

Rainer Blatt. 14-qubit entanglement: Creation and coherence. Physical Review

Letters, 106(13):130506, Jan 2011.

[137] Giovanna Morigi. Cooling atomic motion with quantum interference. Physical

Review A, 67(3):033402, Mar 2003.

[138] Giovanna Morigi, Jurgen Eschner, and Christoph Keitel. Ground state laser

cooling using electromagnetically induced transparency. Physical Review Letters,

85(21):4458–4461, 2000.

[139] A H Myerson, D J Szwer, S C Webster, D T C Allcock, M J Curtis, G Imreh,

J A Sherman, D N Stacey, A M Steane, and D M Lucas. High-fidelity readout of

trapped-ion qubits. Physical Review Letters, 100(20):200502, Jan 2008.

[140] Tomohisa Nagata, Ryo Okamoto, Jeremy L O’Brien, Keiji Sasaki, and Shigeki

Takeuchi. Beating the standard quantum limit with four-entangled photons. Sci-

ence, 316(5825):726–729, Jan 2007.

[141] X R Nie, C F Roos, and D F V James. Theory of cross phase modulation for the

vibrational modes of trapped ions. Physics Letters A, 373(4):422–425, 2009.

[142] M A Nielsen. Optical quantum computation using cluster states. Physical Review

Letters, 93(4):040503, Jan 2004.

[143] Michael A. Nielsen and Isaac L. Chuang. Quantum computation and quantum

information. Cambridge University Press, Jan 2000.

[144] J L O’Brien. Optical quantum computing. Science, 318(5856):1567–1570, 2007.

192


[145] Wolfgang Paul. Electromagnetic traps for charged and neutral particles. Reviews

of Modern Physics, 62:531–540, Jul 1990.

[146] Martin B. Plenio and Shashank Virmani. An introduction to entanglement mea-

sures. Quantum Information and Computation, 7(1):1–51, 2007.

[147] R Raussendorf and H J Briegel. A one-way quantum computer. Physical Review

Letters, 86(22):5188–5191, Jan 2001.

[148] R Raussendorf, D E Browne, and H J Briegel. Measurement-based quantum com-

putation on cluster states. Physical Review A, 68(2):022312, Jan 2003.

[149] Robert Raussendorf and Jim Harrington. Fault-tolerant quantum computation

with high threshold in two dimensions. Physical Review Letters, 98(19):190504,

Jan 2007.

[150] R Reichle, D Leibfried, R B Blakestad, J Britton, J D Jost, E Knill, C Langer,

R Ozeri, S Seidelin, and D J Wineland. Transport dynamics of single ions in

segmented microstructured Paul trap arrays. Fortschritte der Physik, 54(8-10):666–

685, Aug 2006.

[151] M Riebe, H Haffner, C F Roos, W Hansel, J Benhelm, G P T Lancaster, T W

Korber, C Becher, F Schmidt-Kaler, and D F V James. Deterministic quantum

teleportation with atoms. Nature, 429(6993):734–737, 2004.

[152] N. Roch, M. E. Schwartz, F. Motzoi, C. Macklin, R. Vijay, A. W. Eddins, A. N.

Korotkov, K. B. Whaley, M. Sarovar, and I. Siddiqi. Observation of measurement-

induced entanglement and quantum trajectories of remote superconducting qubits.

Physical Review Letters, 112:170501, Jan 2014.

[153] C F Roos, M Chwalla, K Kim, M Riebe, and R Blatt. ‘Designer atoms’ for quantum

metrology. Nature, 443(7109):316–319, 2006.

[154] C F Roos, T Monz, K Kim, M Riebe, H Haffner, D F V James, and R Blatt.

Nonlinear coupling of continuous variables at the single quantum level. Physical

Review A, 77(4):040302, 2008.

[155] M A Rowe, A Ben-Kish, B DeMarco, D Leibfried, V Meyer, J Beall, J Britton,

J Hughes, W M Itano, B Jelenkovic, C Langer, T Rosenband, and D J Wineland.

Transport of quantum states and separation of ions in a dual rf ion trap. Quantum

Inform Compu, 2(4):257–271, Jan 2002.

193


[156] M A Rowe, D Kielpinski, V Meyer, C A Sackett, W M Itano, C Monroe, and D J

Wineland. Experimental violation of a Bell’s inequality with efficient detection.

Nature, 409(6822):791–794, Jan 2001.

[157] K. Saeedi, S. Simmons, J. Z. Salvail, P. Dluhy, H. Riemann, N. V. Abrosimov,

P. Becker, H.-J. Pohl, J. J. L. Morton, and M. L. W. Thewalt. Room-temperature

quantum bit storage exceeding 39 minutes using ionized donors in silicon-28. Sci-

ence, 342(6160):830–833, Nov 2013.

[158] Valerio Scarani, Sofyan Iblisdir, Nicolas Gisin, and Antonio Acin. Quantum cloning.

Reviews of Modern Physics, 77(4):1225, 2005.

[159] S Seidelin, J Chiaverini, R Reichle, J Bollinger, D Leibfried, J Britton, J Wesenberg,

R Blakestad, R Epstein, D Hume, W Itano, J Jost, C Langer, R Ozeri, N Shiga,

and D Wineland. Microfabricated surface-electrode ion trap for scalable quantum

information processing. Physical Review Letters, 96(25):253003, Jun 2006.

[160] A Shamir. How to share a secret. Communications of the ACM, 22(11):612–613,

1979.

[161] P W Shor. Polynomial-time algorithms for prime factorization and discrete loga-

rithms on a quantum computer. SIAM Journal on Computing, 26(5):1484–1509,

Jan 1997.

[162] P W Shor and J Preskill. Simple proof of security of the BB84 quantum key

distribution protocol. Physical Review Letters, 85(2):441–444, Jan 2000.

[163] A Sørensen and K Mølmer. Quantum computation with ions in thermal motion.

Physical Review Letters, 82(9):1971–1974, 1999.

[164] R Stock and D F V James. Scalable, high-speed measurement-based quantum

computer using trapped ions. Physical Review Letters, 102(17):170501, 2009.

[165] X Su, A Tan, X Jia, J Zhang, C Xie, and K Peng. Experimental preparation of

quadripartite cluster and Greenberger-Horne-Zeilinger entangled states for contin-

uous variables. Physical Review Letters, 98(7):070502, 2007.

[166] L Tian and P Zoller. Coupled ion-nanomechanical systems. Physical Review Letters,

93(26):266403, Dec 2004.

[167] W Tittel, H Zbinden, and N Gisin. Experimental demonstration of quantum secret

sharing. Physical Review A, 63(4):042301, 2001.

194


[168] E Torrontegui, S Ibanez, Xi Chen, A Ruschhaupt, D Guery-Odelin, and J G

Muga. Fast atomic transport without vibrational heating. Physical Review A,

83(1):013415, Jan 2011.

[169] C H Tseng, S Somaroo, Y Sharf, E Knill, R Laflamme, T F Havel, and D G Cory.

Quantum simulation with natural decoherence. Physical Review A, 62(3):032309,

Jan 2000.

[170] T Tyc and B C Sanders. How to share a continuous-variable quantum secret by

optical interferometry. Physical Review A, 65(4):042310, Jan 2002.

[171] R Ursin, F Tiefenbacher, T Schmitt-Manderbach, H Weier, T Scheidl, M Linden-

thal, B Blauensteiner, T Jennewein, J Perdigues, and P Trojek. Entanglement-

based quantum communication over 144 km. Nature Physics, 3(7):481–486, 2007.

[172] L Vaidman. Instantaneous measurement of nonlocal variables. Physical Review

Letters, 90(1):010402, 2003.

[173] Peter van Loock, Christian Weedbrook, and Mile Gu. Building Gaussian cluster

states by linear optics. Physical Review A, 76(3):032321, Jan 2007.

[174] G Vidal and R F Werner. Computable measure of entanglement. Physical Review

A, 65(3):032314, Feb 2002.

[175] A Walther, F Ziesel, T Ruster, S T Dawkins, K Ott, M Hettrich, K Singer,

F Schmidt-Kaler, and U Poschinger. Controlling fast transport of cold trapped

ions. Physical Review Letters, 109(8):080501, 2012.

[176] Christian Weedbrook, Stefano Pirandola, Raul Garcıa-Patron, Nicolas Cerf, Tim-

othy Ralph, Jeffrey Shapiro, and Seth Lloyd. Gaussian quantum information.

Reviews of Modern Physics, 84(2):621–669, May 2012.

[177] L F Wei, Yu-Xi Liu, and Franco Nori. Engineering quantum pure states of a

trapped cold ion beyond the Lamb-Dicke limit. Physical Review A, 70(6):063801,

Dec 2004.

[178] G Weihs, T Jennewein, C Simon, H Weinfurter, and A Zeilinger. Violation of

Bell’s inequality under strict Einstein locality conditions. Physical Review Letters,

81(23):5039–5043, Jan 1998.

[179] S. Wiesner. Observation of measurement-induced entanglement and quantum tra-

jectories of remote superconducting qubits. Sigact News, 15:78, 1983.

195


[180] D J Wineland, C Monroe, W M Itano, B E King, D Leibfried, C Myatt, and

C Wood. Trapped-ion quantum simulator. Physica Scripta, T76:147–151, Jan

1998.

[181] D J Wineland, C Monroe, W M Itano, D Leibfried, B E King, and D M Meekhof.

Experimental issues in coherent quantum-state manipulation of trapped atomic

ions. Journal of Research of the National Institute of Standards and Technology,

103(3):259–328, Jan 1998.

[182] David J Wineland. Nobel lecture: Superposition, entanglement, and raising

Schrodinger’s cat. Reviews of Modern Physics, 85(3):1103–1114, Jul 2013.

[183] C Wittig. The Landau-Zener formula. Journal of Physical Chemistry B,

109(17):8428–8430, Jan 2005.

[184] Michael M. Wolf, Geza Giedke, and J. I. Cirac. Extremality of Gaussian quantum

states. Physical Review Letters, 96(8):080502, 2006.

[185] W K Wootters and W H Zurek. A single quantum cannot be cloned. Nature,

299(5886):802–803, Jan 1982.

[186] Ze-Liang Xiang, Sahel Ashhab, J. Q. You, and Franco Nori. Hybrid quantum

circuits: Superconducting circuits interacting with other quantum systems. Reviews

of Modern Physics, 85(2):623, 2013.

[187] Shota Yokoyama, Ryuji Ukai, Seiji Armstrong, Chanond Sornphiphatphong,

Toshiyuki Kaji, Shigenari Suzuki, Jun ichi Yoshikawa, Hidehiro Yonezawa, Nicolas

Menicucci, and Akira Furusawa. Ultra-large-scale continuous-variable cluster states

multiplexed in the time domain. Nature Photonics, 7(12):982–986, 2013.

[188] Li Yu, Robert Griffiths, and Scott Cohen. Fast protocols for local implementation

of bipartite nonlocal unitaries. Physical Review A, 85(1):012304, Jan 2012.

[189] M Yukawa, R Ukai, P van Loock, and A Furusawa. Experimental generation of

four-mode continuous-variable cluster states. Physical Review A, 78(1):012301,

2008.

[190] Jing Zhang and Samuel L Braunstein. Continuous-variable Gaussian analog of

cluster states. Physical Review A, 73(3):032318, Mar 2006.

196

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by Hoi-Kwan Lau - University of Toronto T-Space...Prof. Hoi-Kwong Lo. I really appreciate that they agreed to co-supervise my PhD thesis; this was an adventurous decision as they work