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

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

 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

 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

 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

 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

 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

 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

Scene from Mahabharata

Monogamy Scene from Mahabharata

Is sharing QCs between several parties restricted? Is sharing QCs between several parties restricted?

B A C

B A C Maximal QC

B A C No QC

B A C Maximal QC No QC If A and B have maximal QC, they cannot be quantum correlated, at all with C Ekert, PRL’91; Bennett, Brassard, Mermin, PRL’92 C.H. Bennett, et al., PRA 53, 2046 (’96)

B A C Maximal QC No QC If A and B have maximal QC, they cannot be quantum correlated, at all with C

B A C Highly QC Weakly QC

B A C Highly QC Main ingredient of quantum cryptography Sender Receiver Eavesdropper

B A C High CC Classical correlations do not follow!!

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

B A C Monogamy relation for QC measure

B A C

B A C

B A C

If is monogamous, then B A C Coffman, Kundu, Wootters, PRA 61, (’00)

If is monogamous, then B1B1 A BNBN B2B2 N+1-party case

Squared Concurrence is monogamous B1B1 A BNBN B2B2 CKW,PRA’00; Osborne, Verstraete, PRL ’06

B1B1 A BNBN B2B2 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

Entanglement measures Is it entangled?

Entanglement measures Is it entangled? Qualitative query

Entanglement measures Is it entangled? How much? Is it entangled? How much? Qualitative query Quantitative query

Entanglement measures Is it entangled? How much? Is it entangled? How much? For 2 spin-1/2 particles, Entanglement: well – understood qualitatively, quantitatively For 2 spin-1/2 particles, Entanglement: well – understood qualitatively, quantitatively

1.Ent of Formation, Concurrence 2.Logarithmic Negativity Wootters, PRL 80, 2245 (’98) Vidal, Werner, PRA 65, 2245 (1998)

Concurrence: ’s are the square root of the eigen values of

Concurrence: ’s are the square root of the eigen values of

Concurrence: ’s are the square root of the eigen values of Computable measure

: absolute sum of negative eigenvalues of Partial transpose:

: absolute sum of negative eigenvalues of Partial transpose: Computable measure

1.Quantum Discord 2.Quantum Work-deficit Olliver&Zurek, Henderson & Vedral ’01 Oppenheim. Horodeccy, ASD, Sen ’03

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 Mutual information for 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 produces inequivalent quantities for bipartite quantum states.

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 produces inequivalent quantities for bipartite quantum states. The difference is called 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 Corr Discord

H(X|Y) H(Y|X) The difference is called WD Global work Local Work WD

H(X|Y) H(Y|X) The difference is called WD Global work Local Work WD

H(X|Y) H(Y|X) The difference is called WD Global work Local Work WD

 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

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

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

For tripartite state Does it hold?

For tripartite state Does it hold? a000 + b111  generalized GHZ

For tripartite state Does it hold? a000 + b111  generalized GHZ

For tripartite state Does it hold? c001 + d010 + e100  generalized W

Does it hold? c001 + d010 + e100  generalized W

Does it hold? c001 + d010 + e100  generalized W Prabhu, Pati, ASD, Sen, PRA (R)’12

Quantum discord is nonmonogamous

What is it useful for?

 a000 + b111  generalized GHZ  c001 + d010 + e100  generalized W

 a000 + b111  generalized GHZ  c001 + d010 + e100  generalized W SLOCC

 a000 + b111  generalized GHZ  c001 + d010 + e100  generalized W Monogamy of discord can distinguish them.

 a000 + b111  generalized GHZ  c001 + d010 + e100  generalized W Monogamy of discord can distinguish them. GHZs are monogamous. Ws are polygamous. Monogamy of discord can distinguish them. GHZs are monogamous. Ws are polygamous.

Monogamy of discord test { Gen GHZ, Gen W} Positive port Negative port S

Monogamy of discord test {Gen GHZ, Gen W } >0 <0 Gen GHZ S

Tripartite states Discord Monogamy Monogamy test result Gen GHZ>0Satisfy Gen W< 0Violate GHZ class 90%-->0 10%--<0 Satisfy Violate W class< 0Violate Prabhu, Pati, ASD, Sen, PRA (R) (’12) Giorgi, PRA 84, (’11)

Prabhu, Pati, ASD, Sen, PRA 86, (’12)

Quantum discord is nonmonogamous What is it useful for?

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

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

QC Concurrence (C) EoF (E) Discord (D) C2C2 E2E2 D2D2 Monogamy Polygamy

Theorem: If violates monogamy,  an increasing function s.t Theorem: If violates monogamy,  an increasing function s.t

Theorem: If violates monogamy,  an increasing function s.t Theorem: 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

Salini, Prabhu, ASD, Sen, 2012 Polygamy of WD for W Monogamy of (WD) 5 for W c001 + d010 + e100  generalized W

Salini, Prabhu, ASD, Sen, 2012 Polygamy of WD for W Monogamy of (WD) 5 for W c001 + d010 + e100  generalized W

Salini, Prabhu, ASD, Sen, 2012

Randomly generated 3-qubit states by Haar measure

Salini, Prabhu, ASD, Sen, 2012 Randomly generated 3-qubit states by Haar measure of WD

Theorem: WD is monogamous Discord is monogamous Theorem: WD is monogamous Discord is monogamous For pure 3-party state

Theorem: WD is monogamous Discord is monogamous 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

Randomly generated states:

Concurrence EoF Negativity Log-negativity Discord Work-deficit Concurrence EoF Negativity Log-negativity Discord Work-deficit

Randomly generated states: All QCs become monogamous for large no. of parties 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 nonmonogamous 3 qubits Information theoretic measures r nonmonogamous 3 qubits

Information theoretic measures r nonmonogamous Powers of QCs r monogamous

Information theoretic measures r nonmonogamous Powers of QCs r monogamous Large no of parties enforce monogamy

Information theoretic measures r nonmonogamous Powers of QCs r monogamous Large no of parties enforce monogamy arXiv: arXiv: PRA ’13 PRA ’12