Generating and harnessing photonic entanglement

Generating and harnessing photonic entanglement

Funding:

DEPARTMENT OF PHYSICS,UNIVERSITY OF QUEENSLAND

Rohan DaltonMichael HarveyNathan LangfordTill WeinholdJeremy O’BrienGeoff PrydeAndrew White

Stephen BartlettAggie BranczykMichael BremnerJen DoddAndrew DohertyAlexei GilchristGerard MilburnMichael NielsenTim Ralph

Geoff Pryde

Quantum Technology Lab

Theory Colleagues

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Andrew White

www.quantinfo.org

Stephen BartlettAlexei

Gilchrist

JeremyO’Brien Geoff Pryde

Rohan DaltonAgatha

Brancyzk

Nathan LangfordMichaelHarvey

Gerard Milburn Tim Ralph

TillWeinhold

Generating and harnessing photonic entanglement

Talk outline

1. Qubits • CNOT gate • Quantum process tomography • Generalized quantum measurements with photons

2. Qutrits and Qudits• Gaussian spatial modes• Constructing and measuring

qutrits• Use in quantum bit

commitment

QuickTime™ and aTIFF (Uncompressed) decompressorare needed to see this picture.

www.quantinfo.org

H

V

VHH+iVH -VH+VH -iV VHRDLD-

Poincaré Sphere

Single qubits Single qubit gates

|H ( |H |V / √2

Hadamard gate

|H |H |V

Arbitrary rotation gate

? Two-qubit gates

•

Polarization qubits

C0

C1

T0

T1

C0

C1

T0

T1

CSIGN gate

phaseshift

Basic photonic CNOT

CNOT = HT + CSIGN + HT

HT HT

C0C1T0T1

Ralph, Langford, Bell & White, PRA 65, 062324 (2002) Hofmann & Takeuchi, PRA 66, 024308 (2002)

-1/3

1/3

1/3

CSIGN gate

2-photon CNOT operation

C0C1T0T1

Ralph, Langford, Bell & White, PRA 65, 062324 (2002) Hofmann & Takeuchi, PRA 66, 024308 (2002)

-1/3

1/3

1/3

CSIGN gate

bothreflected

bothtransmitted

2-photon CNOT operation

C0C1T0T1

Ralph, Langford, Bell & White, PRA 65, 062324 (2002) Hofmann & Takeuchi, PRA 66, 024308 (2002)

-1/3

1/3

1/3

CSIGN gate

2-photon CNOT operation

C0C1T0T1HWPHWP

Ralph, Langford, Bell & White, PRA 65, 062324 (2002) Hofmann & Takeuchi, PRA 66, 024308 (2002)

-1/3

1/3

1/3

CNOT gate

Control in Control out

Target in Target out

Polarization 2-photon CNOT

T. C. Ralph, quant-ph/0306190

J. L. Dodd et al., quant-ph/0306081

Concatenating CNOTs

2-photon CNOT in the context of scalable QC

•••

•••QUBITSQUBITSaa

Knill, Laflamme and Milburn, Nature 409, 46 (2001)

LINEAROPTICAL

NETWORK

•••

•••

SINGLEPHOTONSSINGLE PHOTON

DETECTION&

FEEDFORWARD

LOQC = “Linear Optics Quantum Computing”

J. L. O’Brien, G. J. Pryde, et al., Nature 426, 264 (2003)

VHC

VHT

Very stable: insensitive to x-y-z translation

VV,HHCT

Non-classical interference

2-photon CNOT circuit

Measured

Average logical fidelity = Tr [MidealMmeas]/4 = 94 ± 2 %

Ideal

O’Brien, Pryde, et al., Nature 426, 264 (2003) O’Brien, Pryde, et al., PRL 93, 080502 (2004)

Truth table

Populations

Coherences

|C|Tin = |0 -1|1 = |H -V|V

Ideal Measured: Real Measured: ImaginaryFidelity = 92 %

State tomography

€

ψ in ?€

ψout

Decompose into “basis” operations, e.g. rotations

Characterizing the gate itself

IIIXIYIZXIXXXYXZYIYXYYYZZIZXZYZZIIIXIYIZXIXXXYXZYIYXYYYZZIZXZYZZ

For a CNOT:

Any physical process can be written as a completely positive map:

IIIXIYIZXIXXXYXZYIYXYYYZZIZXZYZZIIIXIYIZXIXXXYXZYIYXYYYZZIZXZYZZ

Ideal Measured (Re)

+0.250 - 0.25

+0.34

Quantum process tomography

Measured (Re) Measured (Im)

(II, IX, IY, IZ, XI, XY, XZ, YI, YX, YY, YZ, ZI, ZX, ZY, ZZ)

• Physical interpretation? Change basis

CNOT •

= 87%

Quantum process tomography

Gilchrist, Langford, and Nielsen, quant-ph/0408063 O’Brien, Pryde, et al., PRL 93, 080502 (2004)

• Direct measurement of process fidelity

71 measurements 93 ± 1%

7 ± 1%• Average error rate ≤ C2 = 1-Fp where C is process “closeness”

• Average gate fidelitywhere d is the Hilbert space dimension

95 ± 1%

• Chief source of non-ideal gate operation

Gate measures

1. Measurement outcome is correlated with the signal input

2. The measurement does not alter the value of the measured obsevable

3. Repeated measurement yields the same result - quantum state preparation (QSP) 

Grangier et al. Nature 396, 537

QND and Generalized Measurement

|Controlin

|Targetin = |0

|Controlout

|Targetout

Measure

CN

OT

CNOT gate as a QND device

Non-deterministic: when 1 photon is detected in the meter output the measurement is known to have succeeded

1/3

Experimental scheme for QND

Experimental realization

0.90

0.46

0.47

0.51

• Each compares two probability distributions p and q using the classical fidelity:

F > 85% for all input states

Fidelity measures

1/3

V

Half wave plate

Scheme for generalized measurement

1.0

0.8

0.6

0.4

0.2

0.020151050

HWP angle (deg)

KV

K2 + V

2

Pryde, O’Brien, White, Bartlett & Ralph PRL 92, 190402 (2004)

Most advanced general measurement of a qubit: non-destructive; arbitrary strength; any basis; BUT non-deterministic

Complementarity

(|HH> + |VV>) + |VH> + |HV>)“V”

|V> + |H>

Knill, LaFlamme and Milburn, Nature 409, 46 (2001)Pittman, Jacobs and Franson, PRA, 64, 062311 (2001)

Teleported gates fail by making a Z-measurement

LOQCcluster states

Nielsen, PRL, 93, 040503 (04)Browne & Rudolph,quant-ph/0405157

Z-measurement error correction

AverageFidelity96 ± 3 %

O’Brien, PrydeWhite and Ralph, quant-ph/0408064

Syndromemeasured

but notcorrected

QUTRITS

Polarisation qubits:

What if we want to create photonic qudits?

Gaussian spatial modes:• Infinite dimensional

• Discrete

• Orthogonal

• Can describe any paraxial beam

Encoding information in single photons

Gaussian optical mode:

What about other spot shapes?

… for paraxial beams

Gaussian spatial modes

Non-vortex Mode Families

Hermite-Gauss:(HG)

Ince-Gauss:(IG)

Laguerre-Gauss:(LGN)

rectangular

elliptical

cylindrical

Vortex Mode Families (carrying orbital angular momentum)

Laguerre-Gauss:(LGV)

Gouy phase shift

Siegman, Lasers (1986) Bandres et al., Optics Letters 29, 144 (2004)

…

Gaussian spatial modes

Non-degenerate Qutrit |0=L |1=G |2=R

Mair et al., Nature 412, 313 (2001); Vaziri et al., PRL 89, 240401 (2002) Langford et al., PRL 93, 053601 (2004)

H V

D

R

HG

same Gouy phase shift stationary superposition patterns

Degenerate Qubit

G R

LGV

different Gouy phase shifts evolving superposition patterns

Non-degenerate Qubit

Encoding q. information in spatial modes

Hologram production

Low SpatialFrequency

Diffraction Orders

DistributionEnvelope

Spatial ModeConversion

Medium SpatialFrequencyHigh Spatial Frequency

The holograms - how do they work?

Type-I Spontaneous Parametric Down-conversion (SPDC)

What is the spatial mode quantum state of the photon pairs?

Conceptual Experimental Diagram

Post-select coincident photon pairs using counting electronics.Two-photon QST: all possible pairs of single-photon measurements.Analyse the spatial mode using holograms and single-mode fibres.

Spatial mode quantum state tomography

Uses holograms and single-mode fibres (SMFs):

Analyser extinction efficiency:

Spatial Mode Detector Efficiency

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

5.0 10.0 15.0 20.0

horizontal position of hologram (mm)

fibre coupling efficiency

centred

displaced

theory~1:103

Spatial mode analyzer (SMA)

Spatial mode quantum state tomography

populations

Spatial mode quantum state tomography

coherences

Spatial mode quantum state tomography

coherences

Spatial mode quantum state tomography

Re() Im()

Langford et al., quant-ph/0312072

• EOF < 0.704 • SL = 0.18 • Fψ = 0.88

Non-degenerate Qutrit |0=L |1=G |2=R

Two-qutrit quantum state tomography

• Alice should commit to a message and not be able to change it.• Bob should not be able to decode the message until Alice reveals it.• Quantum bit-commitment with arbitrarily good security is impossible• Qutrits offer the best-known BC security levels, whereas qubits do not!

• Communication between mistrustful parties

• Basis of other protocols, e.g. quantum coin flipping

0

?0

29-39-5

29-39-5

Quantum bit commitment

Step 1: Alice starts with our experimentally measured two-qutrit state.

Uses an entangled, two-part system:(a) the proof and (b) the token

subsystems.Assumption: initial state is only source of imperfection.

A Simulated Purification Protocol

Re() Im()

Spekkens and Rudolph, PRA 65, 012310 (2001) Langford et al., quant-ph/0312072

Quantum bit commitment

Step 2: Alice prepares her chosen logical bit.

Step 3: Alice sends the token to Bob (the commitment).

orthogonal two-qutrit states

non-orthogonal token states

Quantum bit commitment

Step 4: Alice sends the proof subsystem to Bob to complete the BC protocol. He decodes the message with a two-qutrit projective measurement:

Distinguishability of token states limits Alice’s control (trace-distance):

Non-orthogonal token states limit Bob’s possible knowledge gain (fidelity):

Quantum bit commitment

• Fidelity:

inaccessible to bestknown BC protocols

achievable withqubits

qutrits

p = 0.29p = 0.19p = 0.09

= p/3 + (1-p) ideal

classical

impossible

• Orthogonal two-qutrit states result in non-orthogonal reduced token states.• Ideal case provides optimal security, but simulated case still does not!

KEY POINTS

Langford et al., quant-ph/0312072

Quantum bit commitment

• Quantum process tomography of CNOT – fully characterize the process in the 2-qubit space

– high fidelity operation useful for q. info and q. physics tests

• Generalized measurement and QND – non-destructive; arbitrary strength; any basis

• Qutrit entanglement – measured, characterized for use in communications protocols

Conclusions