Measurements of General Quantum Correlations in Nuclear Magnetic Resonance Systems

Measurements of General Quantum Correlations in

Nuclear Magnetic Resonance Systems

Eduardo Ribeiro deAzevedo

São Paulo Brazil

• 75 years• 240 courses• 57.000 undergrad students• ~200 Msc. and PHD programs

UNIVERSITY OF SÃO PAULO - USP

UNIVERSITY OF SÃO PAULO AT SÃO CARLOS

São Carlos City:

250.000 people.5 universities: 1 Federal University (UFSCAR). 1 State Univesity (USP). 3 Private Univesities.

USP at São Carlos:

2 Campi, ~8.000 undergrad students

São Carlos Institute of Physics, USP, Brazilwww.ifsc.usp.br

Al

OC1OC6 - PPV

ETL(ionomer)

ITO

DC V

glassemitted

light

NMR QIP in Brazil

CBPF NMR group: Ivan Oliveira, Alberto Passos, Roberto Sarthour, Jair C. C Freitas

(magnetism and magnetic materials)

IFSC NMR group: Tito Bonagamba, Eduardo R. deAzevedo(Solid-State NMR, MRI)

2002

First experiments done in São Carlos using quadrupolar nuclei 2003

First thesis defence in NMR QIP (Fabio A. Bonk at IFSC) and (Juan Bulnes at CBPF)

2005

Publication of the Book Quantum Information Processing by Elsevier2007

2009 Gather with the quantum information theory group at UFABC – Lucas Celeri and Roberto

Serra.

2010

CBPF NMR spectrometer start to operate.

Hiring of new researchers (Alexandre Souza-CBPF, Diogo Pinto IFSC, João Teles-UFSCAR, Ruben Auccaise - UEPG ) tend to strenght this researche area. 2012

PEOPLE INVOLVED

Isabela AlmeidaRuben AuccaiseAlexandre SouzaIvan S. Oliveira Roberto SarthourTito Bonagamba

Experiments TheoryDiogo S. PintoLucas CéleriRoberto SerraJonas Maziero Felipe FanchiniDavid GirolamiGerardo AdessoF. M. PaulaJ. D. MontealegreA SaguiaMarcelo Sarandy

NMR and the QIP

• NMR is also an excellent test bench for studies on open quantum systems:– efficient implementation and manipulation of the quantum

states (excellent control of the unitary transformations coming from the radiofrequency pulses);

– presence of real environments, which can be described by phase damping and generalized amplitude damping channels;

• Experimental demonstration of QIP procedures, including quantum protocols, algorithms, quantum simulations etc.;• Development of many useful tools for QIP, including quantum protocols, algorithms, dynamic decoupling schemes, among others;

?

Quantum Computation

Entanglement

• In certain schemes of quantum computation where the quantum bits are affected by noise, there seems to be a speed-up over classical scenarios even in the presence of negligibly small or vanishing entanglement.

Knill, E.; Laflamme, R. Power of one bit of quantum information. Physical Review Letters, v. 81, n. 25, p. 5672, 1998.

Datta, A.; Shaji, A. and Caves. C. M. Physical Review Letters 100, p.050502, 2008.

Modi, K., Paterek, T., Son, W., Vedral, V. and Williamson M. Unified View of Quantum and Classical Correlations Physical Review Letters, v. 104, p.080501, 2010.

• A possible explanation for the speed up would be quantum correlations different for entanglement.

How to detect them?

General Quantum Correlations

Other types of correlations

?

Quantum Computation

Entanglement

Merali, Z. Nature, v. 474, p. 24, 2011.

Ollivier, H. & Zurek, W. H. Quantum discord: a measure of the quantumness of correlations.Phys. Rev. Lett. 88, 017901 (2001).

Classification of Quantum and Classical States

All CorrelatedStates

Entangled States

Separable StatesC CQ

Separable

Entangled

ClassicallyCorrelated

A Bij i j

ij

p

A Bij i j

ij

p

, ortonormal basis.

ijij

i j

p i i j j

Classification of Quantum and Classical Two-Qubit States


Entangled States


Separable

Entangled


A Bij i j

ij

p

A Bij i j

ij

p

, ortonormal basis.

ijij

i j

p i i j j

• Bell diagonal states:3

1

14 j j j

j

c

1

3 1 2

3 1 2

1 2 3

1 2 3

1 0 00 1 00 1 0

0 0 1

AB

c c cc c c

c c cc c c

1

2

3

0 00 00 0

cC c

c

Correlation Matrix:

Classification of Quantum and Classical States


Entangled States


Separable

Entangled


A Bij i j

ij

p

A Bij i j

ij

p

, ortonormal basis.

ijij

i j

p i i j j

• Bell diagonal states:3

1

14 j j j

j

c

1

44

3

1

11j

ji

jc

NMR sensitive part of the density matrix

In this sense NMR seems to be the perfect tool for probing quantum correlations of separable states and their interaction with the environment;

– Entropic Discord*: disturbance made in a system when a measurement is applied.

*Ollivier, H.; Zurek, W. Physical Review Letters, v. 88, n. 1, p. 017901, 2002.

Quantum Discord

S(ρA)

S(ρB)S(ρAB)

2( ) ( log )S Tr Von Neumann

Entropy


• Mutual information: : :A B A B A BI S S S


Quantum Discord

S(ρA)

S(ρB)S(ρAB)


Entropy


• Mutual information:

• Classical Correlation:

: :A B A B A BI S S S

: Bj

Q A B A A BJ S S


Quantum Discord

S(ρA)

S(ρB)S(ρAB)


Entropy


• Mutual information:

• Classical Correlation:

• Quantum Discord:

: :A B A B A BI S S S

: Bj

Q A B A A BJ S S

: :maxBj

AB A B Q A BD I J


Quantum Discord

S(ρA)

S(ρB)S(ρAB)


Entropy

– For two-qubits Bell diagonal states*:

*Luo, S. Quantum discord for two-qubit systems. Physical Review A, v. 7, n. 4, p. 042303, 2008.

1 2 3max , ,c c c c

1 2 3 2 1 2 3

1 2 3 2 1 2 3

1 2 3 2 1 2 3

1 2 3 2 1 2 3

2 2

1 1 log 141 log 1

1 log 1

1 log 1

1 1log 1 log 12 2

AB AB ABD I C

c c c c c c

c c c c c c

c c c c c c

c c c c c c

c cc c

Quantum Discord

Probing Quantum CorrelationsWhat is required for probing discord and their degradation upon

interaction with the environment?

• To prepare states with different amounts of QCs .

• To perform a reliable read-out of the final states.

• To have a good description and characterization of the system relaxation.

NMR has all that!!!!

Diogo sets a partnership do study quantum discord by NMR

with Roberto Serra and Lucas Céleri ;

• NMR system.

23 16

1QL Z zH I I I I

Sodium dodecyl sulfate in water forming a lyotropic liquid crystal – 23Na NMR

Anatoly K. Khitrin and B. M. Fung. The Journal of Chemical Physics, 112(16):6963–6965, 2000.

Neeraj Sinha, T. S. Mahesh, K. V. Ramanathan, and Anil Kumar. The Journal of Chemical Physics, 114(10):4415–4420, 2001.

3/2 spins systemSample: Lyotropic Liquid Crystals -Sodium Dodecyl Sulfate (SDS) - Heavy Water (D2O) - Decanol (C10H21OH)

• Strong Modulated Pulase (SMP)*:

1

, ,k

SMP n n nn

U U t

*Fortunato, E.; Pravia, M.; Boulant, N.; Teklemariam, G.; Havel, T.; Cory, D. Design of modulating pulses to implement precise effective hamiltonians for quantum information processing. Journal of Chemical Physics, v. 116, n. 17, p. 7599, 2002. Nelder, J.A.; Mead, R. A simplex-method for function minimization. Computer Journal, v. 7, n. 4, p. 308, 1965.

22

arg

argarg

,,

SMPett

SMPettSMPett

TrTr

TrF

Tools for NMR QIP using quadrupolar Nuclei

Single hard pulse

11 12 13 14

12 22 23 24

13 23 33 34

14 24 34 44

11 12 13 14

12 22 23 24

13 23 33 34

14 24 34 44

11 12 13 14

12 22 23 24

13 23 33 34

14 24 34 44

11 12 13 14

12 22 23 24

13 23 33 34

14 24 34 44

Pure quadrupolar relaxation

Redfield Equations

+

• Generalized Amplitude Damping Channel (GAD):– Longitudinal relaxation (T1)

1 2

3 4

1 0 0, 0 1 0 0

0 01 01 , 00 1

E p E p

E p E p

12 2

L

B

pk T

Two Qubit System

ATt

tCJA ee 12 11 2

BT

ttCJ

B ee 11 11 2

– Global phase damping channel (GPD);

0 1

1 0 0 0 1 0 0 00 1 0 0 0 1 0 0

1 , 0 0 1 0 0 0 1 00 0 0 1 0 0 0 1

E E

01 12

CJ te

11 12 13 14

12 22 23 24

13 23 33 34

14 24 34 44

2 1 2 12 1 2 12 1 2 1

2 1 2 1

11 12 13 14

12 22 23 24

13 23 33 34

14 24 34 44

† †0 0 1 1E E E E

|

randomX

time (ms)

MonotonicalDecay

Different amount of classical and correations in each state

HOWEVER....

• Decoherence Process in Bell-diagonal States:→ Local Phase Damping Channel:

*Maziero, J. and et al. Physical Review A, v. 80, p. 044102, 2009.

3 1 20 0 e 0c c c 3 0 0c 3 10 0 ec c 2ou 0c

Mutual InformationClassical CorrelationEntropic Discord

Time (s)

Sudden-Change Phenomena:

• Decoherence Process in Bell-diagonal States:→ Phase Damping Channel:


3 1 20 0 e 0c c c 3 0 0c 3 10 0 ec c 2ou 0c


Time (s) Time (s)




3 1 20 0 e 0c c c 3 0 0c 3 10 0 ec c 2ou 0c


Time (s) Time (s)Time (s)




3 1 20 0 e 0c c c 3 0 0c 3 10 0 ec c 2ou 0c


Time (s) Time (s)Time (s)


− 2 spins 1/2:2A B

L Z L Z Z zH I S JI S

• Two physical Qubits - NMR representation:

• Generalized Amplitude Damping Channel:– Longitudinal relaxation (T1)

1 2

3 4

1 0 0, 0 1 0 0

0 01 01 , 00 1

E p E p

E p E p

11 12

t Te 12 2

L

B

pk T

Energy exchange between

system and environment

• Phase Damping Channel:- Transversal relaxation (T2):Loss of coherence

without loss of energy

1 2

1 0 1 0, 1

0 1 0 1E E

221 12

tTe

0 0 00 0 010 0 020 0 0

0 00 010 02

0 0

3 1 20 0 e 0c c c

Mutual informationClassical correlationQuantum correlation

3 10 0 ec c 2ou 0c

Mutual informationClassical correlationQuantum correlation

Hilbert-Schmidt distance between the state and the nearest classical state;

C

Sρ

E

D 2

22minG C

D

*Dakic, B.; Vedral, V.; Brukner, C. Necessary and sufficient condition for nonzero quantum discord. Physical Review Letters, v. 105, n. 19, p. 190502, 2010. Girolami, D.; Adesso, G. Observable measure of bipartite quantum correlations. Physical Review Letters, v. 108, n. 15, p. 150403, 2012. Modi, K. and et al. Unified view of quantum and classical correlations. Physical Review Letters, v. 104, n. 8, p. 080501, 2010.

Geometric Discord

Diogo sets a partneship with Gerardo Adesso

3 3 3

1 1 , 1

14 i i i i ij i j

i i i j

x y C

1 1 1 1

12 TrGD S k

• 2 q-bits:

3 3 3

1 1 , 1

14 i i i i ij i j

i i i j

x y C

1 1 1 1

12 TrGD S k

14

t tS xx CC

• 2 q-bits:

3 3 3

1 1 , 1

14 i i i i ij i j

i i i j

x y C

1 1 1 1

12 TrGD S k 22

1

6Tr 2TrTrcos

3 3 3

S SSk

14

t tS xx CC

• Para um sistema de 2 q-bits:

3 3 3

1 1 , 1

14 i i i i ij i j

i i i j

x y C

1 1 1 1

12 TrGD S k 22

1

6Tr 2TrTrcos

3 3 3

S SSk

14

t tS xx CC

• 2 q-bits:

3 2 3322

2arccos 2Tr 9Tr Tr 9Tr3Tr Tr

S S S SS S

Tr 1 1

Tr 1 1i i i

i i i

x

y

Tr 4ij i j i j i jc I I

• Direct Measurement Method:3 3 3

1 1 1

1 1 1 1 14 i i i i ij i j

i i i

x y c

Tr 1 1

Tr 1 1i i i

i i i

x

y

NMR

Observables

Tr 4ij i j i j i jc I I Zero and Double

Quantum Coherences

and anti-phase magnetizations

Convert into a local measurement:

• Direct Measurement Method:3 3 3

1 1 1

1 1 1 1 14 i i i i ij i j

i i i

x y c

Tr 1 1

Tr 1 1i i i

i i i

x

y

NMR

Observables

Tr 4ij i j i j i jc I I

Convert into a local measurement: Tr Tr 1i j i ij

†,, onde i jij ij ij ij A B ijU U U CNOT R

Zero and Double

Quantum Coherences

and anti-phase magnetizations

Θ

j/i 1 2 3

1 0 3π/2 π/2

2 3π/2 π/2 - π/2

3 π/2 - π/2 π/2

1 2 3 x z y

†,, onde i jij ij ij ij A B ijU U U CNOT R

– Negativity of Quantumness (QNA)*:

Minimum amount of entanglement created between the system

and its measurement apparatus in a local measurement;

Geometric measurement (trace norm);

J. D. Montealegre, F. M. Paula, A. Saguia, and M. S. Sarandy, Phys. Rev. A 87, 042115 (2013).

T. Nakano, M. Piani, and G. Adesso, Phys. Rev. A 88, 012117 (2013).





1,

1min 12

AN AB ijB A i j

Q

• Bell diagonal states:







1,

1min 12

AN AB ijB A i j

Q

int

2A

N AB

cQ

3

1

14 j j j

j

c

1

• Bell diagonal states:



• Freezing phenomenon:– Initial state condition*:

– Eg.: c1 = 1, c2 = -0.2, c3 = 0.2 (λ0 = 0, λ1 = 0, λ2 = 0.6, λ3 = 0.4)

*You, B.; Cen, L-X. Physical Review A, v. 86, p. 012102, 2012.

0 3 1 2 0 3 1 2

0 1 2 3 0 1 3 2

e 0

ou e 0

• Freezing phenomenon:– Initial state condition*:

– Eg.: c1 = 1, c2 = -0.2, c3 = 0.2 (λ0 = 0, λ1 = 0, λ2 = 0.6, λ3 = 0.4)

*You, B.; Cen, L-X. Physical Review A, v. 86, p. 012102, 2012.

0 3 1 2 0 3 1 2

0 1 2 3 0 1 3 2

e 0

ou e 0

1 1

2 2

3 3

0

0

0

t

t

c t e c

c t e c

c t c

2 2

2 2

A B

A B

T TT T

Time (s)

2 D

G

DG

1 1

2 2

3 3

1 1 0

1 1 0

1 0

a b

a b

a b a b

c t c

c t c

c t c

1 te 1T

• Generalized Amplitude Damping Channel:

• 2 qubits system represented by 2 coupled spins ½:– Sample: 100 mg of 13C-labeled CHCl3 dissolvedin 0.7 mL CDCl3– Spectrometer:

– Initial State:

H – 500 MHz, T1 = 9 s, T2 = 1.2 s C – 125 MHz, T1 = 25 s, T2 = 0.18 sAcoplamento J – 215.1 Hz

Varian Premium Shielded – 11 T

1 2 30.5, 0.06, 0.24c c c 3 10 0 ec c 2ou 0c

Fidelity = 0.993

• 2 qubits system represented by 2 coupled spins ½:– Sample: 100 mg of 13C-labeled CHCl3 dissolvedin 0.7 mL CDCl3– Spectrometer:

– Initial State:

H – 500 MHz, T1 = 9 s, T2 = 1.2 s C – 125 MHz, T1 = 25 s, T2 = 0.18 sAcoplamento J – 215.1 Hz

Varian Premium Shielded – 11 T

1 2 30.5, 0.06, 0.24c c c 3 10 0 ec c 2ou 0c

– 1º State: 1 2 3 0.2c c c

Fidelity = 0.994

3 1 20 0 e 0c c c

– 2º State: 1 2 30.5, 0.06, 0.24c c c

Fidelity = 0.993

3 10 0 ec c 2ou 0c

• Geometric Discord:

Time (s)

Time (s)

Time (s)

Time (s)

Direct Measurement Tomography Theoretical

• Negativity of Quantumness:

Time (s) Time (s)

Time (s) Time (s)

(Theoretical)

(Theoretical)

(a) Discord (b) Geometric Discord (c) Trace Distance (d) Bures Distance

Preliminary Results

Aaronson, B.; Lo Franco, R.; Adesso, G. Physical Review A, v. 88, p. 012120, 2013.

Freezing Universality

Relaxation Process

Phase Damping (PD)

Generalized Amplitude Damping (GAD)

Decoherence Channels:

• Phase Damping Channel:

- Transversal relaxation (T2):Loss of coherence

without loss of energy

1 2

1 0 1 0, 1

0 1 0 1E E

221 12

tTe

Two Qubit System

*Souza, A.M. and et al. Quantum Information Computation, v. 10, p. 653, 2010.


- Transversal relaxation (T2):

- Global Phase Damping (spin 3/2 system)*:

Loss of coherence without loss of

energy

1 2

1 0 1 0, 1

0 1 0 1E E

221 12

tTe

0 1

1 0 0 0 1 0 0 00 1 0 0 0 1 0 0

1 , 0 0 1 0 0 0 1 00 0 0 1 0 0 0 1

E E

0

1 12

CJ te


Two Qubit System





energy

1 2

1 0 1 0, 1

0 1 0 1E E

221 12

tTe

0 1

1 0 0 0 1 0 0 00 1 0 0 0 1 0 0

1 , 0 0 1 0 0 0 1 00 0 0 1 0 0 0 1

E E

0

1 12

CJ te

11 12 13 14

12 22 23 24

13 23 33 34

14 24 34 44


Two Qubit System





energy

1 2

1 0 1 0, 1

0 1 0 1E E

221 12

tTe

0 1

1 0 0 0 1 0 0 00 1 0 0 0 1 0 0

1 , 0 0 1 0 0 0 1 00 0 0 1 0 0 0 1

E E

0

1 12

CJ te

11 12 13 14

12 22 23 24

13 23 33 34

14 24 34 44


Two Qubit System

Emergence of the Pointer Basis:

S EA

Measurement

Decoherence

Collapse of A in some classical state which is not altered by decoherence!

Pointer Basis

The pointer basis emerges when

classical correlation between S and A

becomes constant!*


Time (s)

• Phase Damping Channel:– 2 spins ½ system

• Generalized Amplitude Damping Channel:– 3/2 spins system– Sample: Lyotropic Liquid Crystals

• Sodium Dodecyl Sulfate (SDS)• Heavy Water (D2O)• Decanol (C10H21OH)

– Spectrometer: Varian Inova – 8 TNa – 92 MHzνQ = 10.4 kHz

A B1

1 2

1T 11.3 ms2CJ

• Differences between representing two qubit systems with two spins 1/2 coupled and one spin 3/2.

• Effects of phase damping and generalized amplitude damping channels.

• Experimental observation of Sudden-change, Freezing, Double Sudden-Change phenomena and the emergence of Pointer Basis.

Conclusion

Acknowledgments

Documents

Measurements of General Quantum Correlations in Nuclear Magnetic Resonance Systems