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?
A
B
C
Maximal QC
B
A
C
Maximal QC
B
A
No QC
C
If A and B have maximal QC,
they cannot be quantum correlated, at all with C
Maximal QC
B
A
No QC
C
Ekert, PRL’91; Bennett, Brassard, Mermin, PRL’92
C.H. Bennett, et al., PRA 53, 2046 (’96)
If A and B have maximal QC,
they cannot be quantum correlated, at all with C
Maximal QC
B
A
No QC
C
Highly QC
B
A
Weakly QC
C
Highly QC
Receiver
B
A
Sender
C
Eavesdropper
Main ingredient of quantum cryptography
High CC
A
High CC
B
C
Classical correlations do not follow!!
Highly QC
A
B
Trade-off between the amount of QCs
between A-B and A-C -- Monogamy
Weakly QC
C
Monogamy relation for QC measure
A
B
C
Monogamy relation for QC measure
A
B
C
Monogamy relation for QC measure
A
B
C
Monogamy relation for QC measure
A
B
C
A
B
If
is monogamous, then
C
Coffman, Kundu, Wootters, PRA 61, 052306 (’00)
A
N+1-party case
B1
B2
If
BN
is monogamous, then
CKW,PRA’00; Osborne, Verstraete, PRL ’06
A
B1
B2
Squared Concurrence is monogamous
BN
A
BMultiparty
1
Properties by bipartite measures
B2
BN


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
Is it entangled?
Qualitative query
Is it entangled?
Qualitative query
Is it entangled?
How much?
Quantitative query
For 2 spin-1/2 particles,
Is it entangled?
Entanglement: well – understood
How much?
qualitatively, quantitatively
1. Ent of Formation, Concurrence
Wootters, PRL 80, 2245 (’98)
2. Logarithmic Negativity
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
: absolute sum of negative eigenvalues of
Partial transpose:
: absolute sum of negative eigenvalues of
Partial transpose:
1. Quantum Discord
Olliver&Zurek, Henderson & Vedral ’01
2. Quantum Work-deficit
Oppenheim. Horodeccy, ASD, Sen ’03
H = Shannon Entropy
H(Y)
H(X)
Mutual information
H(X|Y)
for classical random variables.
H(Y|X)
H(X,Y)
or
H(X) + H(Y) – H(X,Y)
H(X) – H(X|Y)
H(Y)
H(X)
Quantizing them produces
inequivalent quantities
H(X|Y)
H(Y|X)
for bipartite quantum states.
H(X,Y)
or
H(X) + H(Y) – H(X,Y)
H(X) – H(X|Y)
H(Y)
H(X)
Quantizing them produces
inequivalent quantities
H(X|Y)
H(Y|X)
for bipartite quantum states.
The difference is called Discord.
H(X,Y)
or
H(X) + H(Y) – H(X,Y)
H(X) – H(X|Y)
H(Y)
H(X)
Discord
Total Corr
H(X|Y)
Classical Corr
H(Y|X)
The difference is called Discord.
H(X,Y)
or
S(X) + S(Y) – S(X,Y)
S(X) – S(X|Y)
WD
Global work
H(X|Y)
Local Work
H(Y|X)
The difference is called WD
WD
Global work
H(X|Y)
Local Work
H(Y|X)
The difference is called WD
WD
Global work
H(X|Y)
Local Work
H(Y|X)
The difference is called 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 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

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.
{Gen GHZ, Gen W}
S
Positive port
Monogamy of
discord test
Negative port
Gen GHZ
{Gen GHZ, Gen W}
S
Monogamy of
discord test
Tripartite
states
Discord
Monogamy
Monogamy
test result
Gen GHZ
>0
Satisfy
Gen W
<0
Violate
90%-->0
10%--<0
Satisfy
Violate
<0
Violate
GHZ class
W class
Prabhu, Pati, ASD, Sen, PRA 86 040102(R) (’12)
Giorgi, PRA 84, 054301 (’11)
Prabhu, Pati, ASD, Sen, PRA 86, 052337 (’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 EoF Discord
(C)
Monogamy
Polygamy
(E)
(D)
C2
E2
D2
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
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
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
Salini, Prabhu, ASD, Sen, 2012
Theorem:
For pure 3-party state
WD is monogamous
Discord is monogamous
Theorem:
For pure 3-party state
WD is monogamous
Discord is monogamous
This is because

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
Asutosh’s poster
Randomly generated states: All QCs become monogamous
for large no. of parties
Asutosh, Prabhu, ASD, Sen, arXiv: 1312:6640
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:1312.6640
arXiv:1206.4029
PRA ’13
PRA ’12