monogamy of quantum correlations
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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. Outline. - PowerPoint PPT PresentationTRANSCRIPT
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Monogamy of Quantum Correlations
Aditi Sen (De)Harish-Chandra Research Institute, India
![Page 2: Monogamy of Quantum Correlations](https://reader030.vdocument.in/reader030/viewer/2022033022/56816718550346895ddb8a07/html5/thumbnails/2.jpg)
Monogamy of Quantum Correlations
Aditi Sen (De)Harish-Chandra Research Institute, India
Co-workers: Asutosh, Salini, Arun, Prabhu, Ujjwal
![Page 3: Monogamy of Quantum Correlations](https://reader030.vdocument.in/reader030/viewer/2022033022/56816718550346895ddb8a07/html5/thumbnails/3.jpg)
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
![Page 4: Monogamy of Quantum Correlations](https://reader030.vdocument.in/reader030/viewer/2022033022/56816718550346895ddb8a07/html5/thumbnails/4.jpg)
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
![Page 5: Monogamy of Quantum Correlations](https://reader030.vdocument.in/reader030/viewer/2022033022/56816718550346895ddb8a07/html5/thumbnails/5.jpg)
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
![Page 6: Monogamy of Quantum Correlations](https://reader030.vdocument.in/reader030/viewer/2022033022/56816718550346895ddb8a07/html5/thumbnails/6.jpg)
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
![Page 7: Monogamy of Quantum Correlations](https://reader030.vdocument.in/reader030/viewer/2022033022/56816718550346895ddb8a07/html5/thumbnails/7.jpg)
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
![Page 8: Monogamy of Quantum Correlations](https://reader030.vdocument.in/reader030/viewer/2022033022/56816718550346895ddb8a07/html5/thumbnails/8.jpg)
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
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What is monogamy of QC?
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
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What is monogamy?
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What is monogamy?
Scene from Mahabharata
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What is monogamy?
MonogamyScene from Mahabharata
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Is sharing QCs between several parties restricted?
What is monogamy?
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What is monogamy?
B
A
C
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What is monogamy?
B
A
C
Maximal QC
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What is monogamy?
B
A
C
Maximal QCNo QC
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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)
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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
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What is monogamy?
B
A
C
Highly QCWeakly QC
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What is monogamy?
B
A
C
Highly QC
Main ingredient of quantum cryptography
Sender
Receiver
Eavesdropper
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What is monogamy?
B
A
C
High CCHigh CC
Classical correlations do not follow!!
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What is monogamy?
B
A
C
Highly QCWeakly QC
Trade-off between the amount of QCs between A-B and A-C -- Monogamy
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Monogamy-Quantification
B
A
C
Monogamy relation for QC measure
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Monogamy-Quantification
B
A
C
Monogamy relation for QC measure
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Monogamy-Quantification
B
A
C
Monogamy relation for QC measure
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Monogamy-Quantification
B
A
C
Monogamy relation for QC measure
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If is monogamous, then
Monogamy-Quantification
B
A
C
Coffman, Kundu, Wootters, PRA 61, 052306 (’00)
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If is monogamous, then
Monogamy-Quantification
B1
A
BNB2
N+1-party case
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Squared Concurrence is monogamous
Monogamy of concurrence
B1
A
BNB2
CKW,PRA’00; Osborne, Verstraete, PRL ’06
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Monogamy-Quantification
B1
A
BNB2
Multiparty Properties by bipartite measures
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What is monogamy of quantum correlations?
QC measures 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
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Entanglement measures
Is it entangled?
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Entanglement measures
Is it entangled?Qualitative query
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Entanglement measures
Is it entangled?How much?
Qualitative query
Quantitative query
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Entanglement measures
Is it entangled?How much?
For 2 spin-1/2 particles, Entanglement: well – understood
qualitatively, quantitatively
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Entanglement Measures
1. Ent of Formation, Concurrence
2. Logarithmic Negativity
Wootters, PRL 80, 2245 (’98)
Vidal, Werner, PRA 65, 2245 (1998)
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Ent of FormationConcurrence:
’s are the square root of the eigen values of
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Ent of FormationConcurrence:
’s are the square root of the eigen values of
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Ent of FormationConcurrence:
’s are the square root of the eigen values of
Computable measure
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: absolute sum of negative eigenvalues of
Partial transpose:
Log Neg
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: absolute sum of negative eigenvalues of
Partial transpose:
Log Neg
Computable measure
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Information-theoretic Measures
1. Quantum Discord
2. Quantum Work-deficit
Olliver&Zurek, Henderson & Vedral ’01
Oppenheim. Horodeccy, ASD, Sen ’03
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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
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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
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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
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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
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Quantum Work-deficit
H(X|Y) H(Y|X)The difference is called WD
Global work Local WorkWD
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Quantum Work-deficit
H(X|Y) H(Y|X)The difference is called WD
Global work Local WorkWD
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Quantum Work-deficit
H(X|Y) H(Y|X)The difference is called WD
Global work Local WorkWD
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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
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Does quantum discord satisfy monogamy relation?
Does the sharing of quantum discord follow the same broad guidelines that are followed by entanglement?
Questions
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Does quantum discord satisfy monogamy relation?
Does the sharing of quantum discord follow the same broad guidelines that are followed by entanglement?
Questions
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For tripartite state
Monogamy of Q Discord
Does it hold?
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For tripartite state
Monogamy of Q Discord
Does it hold?
a000 + b111 generalized GHZ
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For tripartite state
Monogamy of Q Discord
Does it hold?
a000 + b111 generalized GHZ
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For tripartite state
Monogamy of Q Discord
Does it hold?
c001 + d010 + e100 generalized W
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Monogamy of Q Discord
Does it hold?
c001 + d010 + e100 generalized W
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Monogamy of Q Discord
Does it hold?
c001 + d010 + e100 generalized W
Prabhu, Pati, ASD, Sen, PRA 86 040102(R)’12
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Quantum discord is nonmonogamous
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Quantum discord is nonmonogamous
What is it useful for?
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a000 + b111 generalized GHZ
c001 + d010 + e100 generalized W
GHZ vs W
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a000 + b111 generalized GHZ
c001 + d010 + e100 generalized W
GHZ vs WSLOCC
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a000 + b111 generalized GHZ
c001 + d010 + e100 generalized W
GHZ vs W
Monogamy of discord can distinguish them.
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a000 + b111 generalized GHZ
c001 + d010 + e100 generalized W
GHZ vs W
Monogamy of discord can distinguish them.GHZs are monogamous.
Ws are polygamous.
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GHZ vs W
Monogamy of discord test
{Gen GHZ, Gen W}Positive port
Negative portS
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GHZ vs W
Monogamy of discord test
{Gen GHZ, Gen W}>0
<0
Gen GHZ
S
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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)
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Monogamy of QD
Prabhu, Pati, ASD, Sen, PRA 86, 052337 (’12)
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Quantum discord is nonmonogamous
What is it useful for?
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NMR
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NMR
Monogamy of discord helps to distinguish two phases in many-body system
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NMR
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NMR
Koteswar Rao, Katiyar, Mahesh, ASD, Sen, Kumar, PRA 2013
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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
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Powers of QCQC Concurrence
(C)EoF(E)
Discord(D)
C2 E2 D2
Monogamy
Polygamy
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Powers of QCTheorem:
If violates monogamy, an increasing function s.t
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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
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Powers of WD: gen W state
Salini, Prabhu, ASD, Sen, 2012
Polygamy of WD for W Monogamy of (WD)5 for W
c001 + d010 + e100 generalized W
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Powers of WD: gen W state
Salini, Prabhu, ASD, Sen, 2012
Polygamy of WD for W Monogamy of (WD)5 for W
c001 + d010 + e100 generalized W
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Powers of WD: Monogamous
Salini, Prabhu, ASD, Sen, 2012
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Powers of WD: Monogamous
Salini, Prabhu, ASD, Sen, 2012
Randomly generated 3-qubit states by Haar measure
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Powers of WD: Monogamous
Salini, Prabhu, ASD, Sen, 2012
Randomly generated 3-qubit states by Haar measure
of WD
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Monogamy of WD
Theorem:
WD is monogamous Discord is monogamousFor pure 3-party state
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Monogamy of WD
Theorem:
WD is monogamous Discord is monogamous
This is because
For pure 3-party state
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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 large no of
parties
Outline
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Large no. of parties help to get monogamy
Randomly generated states:
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Large no. of parties help to get monogamy
Concurrence
EoF
Negativity
Log-negativity
Discord
Work-deficit
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Large no. of parties help to get monogamy
Randomly generated states: All QCs become monogamous for large no. of parties
Asutosh’s poster
Asutosh, Prabhu, ASD, Sen, arXiv: 1312:6640
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Information theoretic measures r nonmonogamous3 qubits
Conclusion
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Information theoretic measures r nonmonogamous
Powers of QCs r monogamous
Conclusion
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Information theoretic measures r nonmonogamous
Powers of QCs r monogamous
Large no of parties enforce monogamy
Conclusion
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Information theoretic measures r nonmonogamous
Powers of QCs r monogamous
Large no of parties enforce monogamy
arXiv:1312.6640arXiv:1206.4029PRA ’13PRA ’12
Conclusion
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