Monogamy of Quantum Correlations

Monogamy of Quantum Correlations

Aditi Sen (De)Harish-Chandra Research Institute, India

Monogamy of Quantum Correlations

Aditi Sen (De)Harish-Chandra Research Institute, India

Co-workers: Asutosh, Salini, Arun, Prabhu, Ujjwal

What is monogamy of quantum correlations?

Quantum Correlations Entanglement-Separability paradigm Information-theoretic paradigm

Info-theoretic measures: nonmonogamous (3-qubits)

R quantum correlations monogamous? Powers of QC r monogamous QC r monogamous for higher no of parties

Outline

What is monogamy of QC?




Outline

What is monogamy?

What is monogamy?

Scene from Mahabharata

What is monogamy?

MonogamyScene from Mahabharata

Is sharing QCs between several parties restricted?

What is monogamy?

What is monogamy?

B

A

C

What is monogamy?

B

A

C

Maximal QC

What is monogamy?

B

A

C

Maximal QCNo QC

What is monogamy?

B

A

C

Maximal QCNo QC

If A and B have maximal QC,they cannot be quantum correlated, at all with C

Ekert, PRL’91; Bennett, Brassard, Mermin, PRL’92C.H. Bennett, et al., PRA 53, 2046 (’96)

What is monogamy?

B

A

C

Maximal QCNo QC

If A and B have maximal QC,they cannot be quantum correlated, at all with C

What is monogamy?

B

A

C

Highly QCWeakly QC

What is monogamy?

B

A

C

Highly QC

Main ingredient of quantum cryptography

Sender

Receiver

Eavesdropper

What is monogamy?

B

A

C

High CCHigh CC

Classical correlations do not follow!!

What is monogamy?

B

A

C

Highly QCWeakly QC

Trade-off between the amount of QCs between A-B and A-C -- Monogamy

Monogamy-Quantification

B

A

C

Monogamy relation for QC measure

If is monogamous, then


B

A

C

Coffman, Kundu, Wootters, PRA 61, 052306 (’00)

If is monogamous, then


B1

A

BNB2

N+1-party case

Squared Concurrence is monogamous

Monogamy of concurrence

B1

A

BNB2

CKW,PRA’00; Osborne, Verstraete, PRL ’06


B1

A

BNB2

Multiparty Properties by bipartite measures


QC measures Entanglement-Separability

paradigm Information-theoretic paradigm



Outline

Entanglement measures

Is it entangled?


Is it entangled?Qualitative query


Is it entangled?How much?

Qualitative query

Quantitative query


Is it entangled?How much?

For 2 spin-1/2 particles, Entanglement: well – understood

qualitatively, quantitatively

Entanglement Measures

1. Ent of Formation, Concurrence

2. Logarithmic Negativity

Wootters, PRL 80, 2245 (’98)

Vidal, Werner, PRA 65, 2245 (1998)

Ent of FormationConcurrence:

’s are the square root of the eigen values of

Ent of FormationConcurrence:

’s are the square root of the eigen values of

Computable measure

: absolute sum of negative eigenvalues of

Partial transpose:

Log Neg

: absolute sum of negative eigenvalues of

Partial transpose:

Log Neg

Computable measure

Information-theoretic Measures

1. Quantum Discord

2. Quantum Work-deficit

Olliver&Zurek, Henderson & Vedral ’01

Oppenheim. Horodeccy, ASD, Sen ’03

H(X|Y)

Quantum discordH(X) H(Y)

H(Y|X)

H(X,Y)H(X) + H(Y) – H(X,Y)

H(X) – H(X|Y)or

Mutual informationfor classical random variables.

H = Shannon Entropy

H(X|Y)

H(X) H(Y)

H(Y|X)

H(X,Y)H(X) + H(Y) – H(X,Y)

H(X) – H(X|Y)or

Quantizing them producesinequivalent quantities

for bipartite quantum states.

Quantum discord

H(X|Y)

H(X) H(Y)

H(Y|X)

H(X,Y)H(X) + H(Y) – H(X,Y)

H(X) – H(X|Y)or

Quantizing them producesinequivalent quantities

for bipartite quantum states.

The difference is called Discord.

Quantum discord

H(X|Y)

H(X) H(Y)

H(Y|X)

H(X,Y)S(X) + S(Y) – S(X,Y)

S(X) – S(X|Y)or

The difference is called Discord.

Total Corr Classical CorrDiscord

Quantum discord

Quantum Work-deficit

H(X|Y) H(Y|X)The difference is called WD

Global work Local WorkWD



Info-theoretic measures: nonmonogamous

(3-qubits)


Outline

Does quantum discord satisfy monogamy relation?

Does the sharing of quantum discord follow the same broad guidelines that are followed by entanglement?

Questions

For tripartite state

Monogamy of Q Discord

Does it hold?



Does it hold?

a000 + b111 generalized GHZ



Does it hold?

c001 + d010 + e100 generalized W


Does it hold?



Does it hold?


Prabhu, Pati, ASD, Sen, PRA 86 040102(R)’12

Quantum discord is nonmonogamous


What is it useful for?



GHZ vs W



GHZ vs WSLOCC



GHZ vs W

Monogamy of discord can distinguish them.



GHZ vs W

Monogamy of discord can distinguish them.GHZs are monogamous.

Ws are polygamous.

GHZ vs W

Monogamy of discord test

{Gen GHZ, Gen W}Positive port

Negative portS

GHZ vs W

Monogamy of discord test

{Gen GHZ, Gen W}>0

<0

Gen GHZ

S

Tripartite states

Discord Monogam

y

Monogamy test result

Gen GHZ >0 Satisfy

Gen W < 0 Violate

GHZ class 90%-->010%--<0

SatisfyViolate

W class < 0 Violate

Results for Discord monogamy test

Prabhu, Pati, ASD, Sen, PRA 86 040102(R) (’12)Giorgi, PRA 84, 054301 (’11)

Monogamy of QD

Prabhu, Pati, ASD, Sen, PRA 86, 052337 (’12)


What is it useful for?

NMR

Monogamy of discord helps to distinguish two phases in many-body system

NMR

Koteswar Rao, Katiyar, Mahesh, ASD, Sen, Kumar, PRA 2013





Outline

Powers of QCQC Concurrence

(C)EoF(E)

Discord(D)

C2 E2 D2

Monogamy

Polygamy

Powers of QCTheorem:

If violates monogamy, an increasing function s.t

Powers of QCTheorem:

If violates monogamy, an increasing function s.t

provided is monotonically decreasing under discarding system& invariance under discarding systems occurs only for monogamy

satisfying states

Powers of WD: gen W state

Salini, Prabhu, ASD, Sen, 2012

Polygamy of WD for W Monogamy of (WD)5 for W


Powers of WD: Monogamous




Randomly generated 3-qubit states by Haar measure



Randomly generated 3-qubit states by Haar measure

of WD

Monogamy of WD

Theorem:

WD is monogamous Discord is monogamousFor pure 3-party state

Monogamy of WD

Theorem:

WD is monogamous Discord is monogamous

This is because

For pure 3-party state




R quantum correlations monogamous? Powers of QC r monogamous QC r monogamous for large no of

parties

Outline

Large no. of parties help to get monogamy

Randomly generated states:


Concurrence

EoF

Negativity

Log-negativity

Discord

Work-deficit


Randomly generated states: All QCs become monogamous for large no. of parties

Asutosh’s poster

Asutosh, Prabhu, ASD, Sen, arXiv: 1312:6640

Information theoretic measures r nonmonogamous3 qubits

Conclusion

Information theoretic measures r nonmonogamous

Powers of QCs r monogamous

Conclusion



Large no of parties enforce monogamy

Conclusion



Large no of parties enforce monogamy

arXiv:1312.6640arXiv:1206.4029PRA ’13PRA ’12

Conclusion

Monogamy of Quantum Correlations

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