Monogamy of Quantum Correlations
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Transcript of 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 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 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 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 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 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?
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?
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
Monogamy-Quantification
B
A
C
Monogamy relation for QC measure
Monogamy-Quantification
B
A
C
Monogamy relation for QC measure
Monogamy-Quantification
B
A
C
Monogamy relation for QC measure
If is monogamous, then
Monogamy-Quantification
B
A
C
Coffman, Kundu, Wootters, PRA 61, 052306 (’00)
If is monogamous, then
Monogamy-Quantification
B1
A
BNB2
N+1-party case
Squared Concurrence is monogamous
Monogamy of concurrence
B1
A
BNB2
CKW,PRA’00; Osborne, Verstraete, PRL ’06
Monogamy-Quantification
B1
A
BNB2
Multiparty Properties by bipartite measures
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
Entanglement measures
Is it entangled?
Entanglement measures
Is it entangled?Qualitative query
Entanglement measures
Is it entangled?How much?
Qualitative query
Quantitative query
Entanglement measures
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
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
Quantum Work-deficit
H(X|Y) H(Y|X)The difference is called WD
Global work Local WorkWD
Quantum Work-deficit
H(X|Y) H(Y|X)The difference is called WD
Global work Local WorkWD
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
Does quantum discord satisfy monogamy relation?
Does the sharing of quantum discord follow the same broad guidelines that are followed by entanglement?
Questions
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?
For tripartite state
Monogamy of Q Discord
Does it hold?
a000 + b111 generalized GHZ
For tripartite state
Monogamy of Q Discord
Does it hold?
a000 + b111 generalized GHZ
For tripartite state
Monogamy of Q Discord
Does it hold?
c001 + d010 + e100 generalized W
Monogamy of Q Discord
Does it hold?
c001 + d010 + e100 generalized W
Monogamy of Q Discord
Does it hold?
c001 + d010 + e100 generalized W
Prabhu, Pati, ASD, Sen, PRA 86 040102(R)’12
Quantum discord is nonmonogamous
Quantum discord is nonmonogamous
What is it useful for?
a000 + b111 generalized GHZ
c001 + d010 + e100 generalized W
GHZ vs W
a000 + b111 generalized GHZ
c001 + d010 + e100 generalized W
GHZ vs WSLOCC
a000 + b111 generalized GHZ
c001 + d010 + e100 generalized W
GHZ vs W
Monogamy of discord can distinguish them.
a000 + b111 generalized GHZ
c001 + d010 + e100 generalized W
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)
Quantum discord is nonmonogamous
What is it useful for?
NMR
NMR
Monogamy of discord helps to distinguish two phases in many-body system
NMR
NMR
Koteswar Rao, Katiyar, Mahesh, ASD, Sen, Kumar, PRA 2013
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
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
c001 + d010 + e100 generalized W
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
Powers of WD: Monogamous
Salini, Prabhu, ASD, Sen, 2012
Powers of WD: Monogamous
Salini, Prabhu, ASD, Sen, 2012
Randomly generated 3-qubit states by Haar measure
Powers of WD: Monogamous
Salini, Prabhu, ASD, Sen, 2012
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
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
Large no. of parties help to get monogamy
Randomly generated states:
Large no. of parties help to get monogamy
Concurrence
EoF
Negativity
Log-negativity
Discord
Work-deficit
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
Information theoretic measures r nonmonogamous3 qubits
Conclusion
Information theoretic measures r nonmonogamous
Powers of QCs r monogamous
Conclusion
Information theoretic measures r nonmonogamous
Powers of QCs r monogamous
Large no of parties enforce monogamy
Conclusion
Information theoretic measures r nonmonogamous
Powers of QCs r monogamous
Large no of parties enforce monogamy
arXiv:1312.6640arXiv:1206.4029PRA ’13PRA ’12
Conclusion