Constrained Quantum CorrelationsConstrained …confqic/qipa15/talks/amit.pdfConstrained Quantum CorrelationsConstrained Quantum Correlations A user-friendly computational scheme Amit

Constrained Quantum CorrelationsConstrained Quantum CorrelationsA user-friendly computational scheme

Amit Kumar PalHarish-Chandra Research Institute, Allahabad, INDIA

QIPA - 2015

CollaboratorsTitas Chanda

Tamoghna DasDebasis SadhukhanSudipto Singha Roy

Asutosh KumarAditi Sen(De)

Ujjwal Sen

PRA 92, 062301 (2015)PRA 91, 062119 (2015)

Quantum correlations beyond entanglement

Several measures available... discord & “discord-like”

1. Quantum discord2. Quantum work deficit3. Several geometric measures... Review by Modi et. al., RMP (2012)

Use as resource? – Debatable?

Quantum correlations beyond entanglement

Several measures available... discord & “discord-like”

1. Quantum discord2. Quantum work deficit3. Several geometric measures... Review by Modi et. al., RMP (2012)

Use as resource? – Debatable?

Quantum discord

∼ Difference between two inequivalent definitions of quantum mutual information

I(ρAB) = S(ρA) + S(ρB)− S(ρAB)

J→(ρAB) = S(ρB)− S(ρB|ρA)S (ρ) = −Tr[ρ log2 ρ]← von Neumann entropyS(ρB|ρA) =

∑k pkS(ρk

AB)← Quantum conditional entropy

D(ρAB) = minSC={ΠA

k }

{S(ρA)− S(ρAB) +

∑k

pkS(ρk

AB

)}

∼ Involves minimization over a complete set of projective measurements SC = {ΠAk }

Ollivier & Zurek, PRL (2001); Henderson & Vedral, J. Phys. A (2001)

.., but how to compute discord?

Optimization over {ΠAk } for general two-qubit states

Two real parameters, θ, φ (0 ≤ θ ≤ π, 0 ≤ φ < 2π)← Bloch sphere

Analytical calculation of discord: Bell-diagonal states onlyLuo, PRA (2008)

ρAB =14

[IA ⊗ IB +∑

α=x,y,z

cαασαA ⊗ σαB ]

Discord of more general two-qubit mixed stateDespite several attempts, only numerical results so far...

Computing quantum discord in NP-completeRunning time grows exponentially with dimension of the Hilbert space

Huang, NJP (2014)

Higher dimensional systems: extremely difficult!





ρAB =14

[IA ⊗ IB +∑

α=x,y,z




Huang, NJP (2014)






ρAB =14

[IA ⊗ IB +∑

α=x,y,z




Huang, NJP (2014)






ρAB =14

[IA ⊗ IB +∑

α=x,y,z




Huang, NJP (2014)






ρAB =14

[IA ⊗ IB +∑

α=x,y,z




Huang, NJP (2014)






ρAB =14

[IA ⊗ IB +∑

α=x,y,z




Huang, NJP (2014)






ρAB =14

[IA ⊗ IB +∑

α=x,y,z




Huang, NJP (2014)


How about a constrained optimization?

Restriction over the set of measurement: “Earmarked” setSE ⊆ SC

Dc = minSE

[I(ρAB)− J→(ρAB)] D = minSC

[I(ρAB)− J→(ρAB)]

n sets of projection measurements in SE

n→∞ may/may not imply SE → SC, i.e., Dc → D

Dc ≥ D −→ εn = Dc − D ←− “Voluntary error”

εn = 0 ←− “Exceptional states”

How about a constrained optimization?Restriction over the set of measurement: “Earmarked” set

SE ⊆ SC

Dc = minSE








SE ⊆ SC

Dc = minSE


D = minSC







SE ⊆ SC

Dc = minSE








SE ⊆ SC

Dc = minSE








SE ⊆ SC

Dc = minSE








SE ⊆ SC

Dc = minSE







n points

εrn =

∫εr

nPrn(εr

n)dεrn

εr∞← “Average asymptotic error”

Two-qubit mixed states of different ranks, r

εrn = εr

∞ + κn−τ

For r = 2τ ≈ 1.92κ ≈ 0.177εr=2∞ ≈ 0.121

For r = 4τ ≈ 1.93κ ≈ 0.145εr=4∞ ≈ 0.078

Better for higher rank

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

5 10 15 20 25 30 35 40 45 50

– εr=

2n −

– ε

r=

2∞

n

QD-numerical

QD-fitted

10-4

10-3

10-2

10-1

2 5 10 20 50 1

n points

εrn =

∫εr

nPrn(εr

n)dεrn



εrn = εr

∞ + κn−τ

For r = 2τ ≈ 1.92κ ≈ 0.177εr=2∞ ≈ 0.121

For r = 4τ ≈ 1.93κ ≈ 0.145εr=4∞ ≈ 0.078


0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

5 10 15 20 25 30 35 40 45 50

– εr=

2n −

– ε

r=

2∞

n

QD-numerical

QD-fitted

10-4

10-3

10-2

10-1

2 5 10 20 50 1

n points

εrn =

∫εr

nPrn(εr

n)dεrn



εrn = εr

∞ + κn−τ

For r = 2τ ≈ 1.92κ ≈ 0.177εr=2∞ ≈ 0.121

For r = 4τ ≈ 1.93κ ≈ 0.145εr=4∞ ≈ 0.078


0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

5 10 15 20 25 30 35 40 45 50

– εr=

2n −

– ε

r=

2∞

n

QD-numerical

QD-fitted

10-4

10-3

10-2

10-1

2 5 10 20 50 1

n points

εrn =

∫εr

nPrn(εr

n)dεrn



εrn = εr

∞ + κn−τ

For r = 2τ ≈ 1.92κ ≈ 0.177εr=2∞ ≈ 0.121

For r = 4τ ≈ 1.93κ ≈ 0.145εr=4∞ ≈ 0.078


0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

5 10 15 20 25 30 35 40 45 50

– εr=

2n −

– ε

r=

2∞

n

QD-numerical

QD-fitted

10-4

10-3

10-2

10-1

2 5 10 20 50 1

n =∑n2

i=1 ni1

0.04

0.06

0.08

0.1

0.12

0.14

5 10 15 20 25 30

– εnr

n2

Rank 2

Rank 3, NPPT

Rank 3, PPT

Rank 4, NPPT

Rank 4, PPT

n1 = 10

n = n1 × n2

for QD

2 4 6 8 10 12 14

n1

2

4

6

8

10

12

14

n2

0

0.05

0.1

0.15

0.2

0.25

A

Rank 2

(a)

2 4 6 8 10 12 14

n1

2

4

6

8

10

12

14

n2

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

A

Rank 4, NPPT

(b)

2 4 6 8 10 12 14

n1

2

4

6

8

10

12

14

n2

0

0.02

0.04

0.06

0.08

0.1

0.12

A

Rank 4, PPT

εrn ≤ 10−3 outside boundary

Faster for higher rank, PPT states

n =∑n2

i=1 ni1

0.04

0.06

0.08

0.1

0.12

0.14

5 10 15 20 25 30

– εnr

n2

Rank 2

Rank 3, NPPT

Rank 3, PPT

Rank 4, NPPT

Rank 4, PPT

n1 = 10

n = n1 × n2

for QD

2 4 6 8 10 12 14

n1

2

4

6

8

10

12

14

n2

0

0.05

0.1

0.15

0.2

0.25

A

Rank 2

(a)

2 4 6 8 10 12 14

n1

2

4

6

8

10

12

14

n2

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

A

Rank 4, NPPT

(b)

2 4 6 8 10 12 14

n1

2

4

6

8

10

12

14

n2

0

0.02

0.04

0.06

0.08

0.1

0.12

A

Rank 4, PPT



n =∑n2

i=1 ni1

0.04

0.06

0.08

0.1

0.12

0.14

5 10 15 20 25 30

– εnr

n2

Rank 2

Rank 3, NPPT

Rank 3, PPT

Rank 4, NPPT

Rank 4, PPT

n1 = 10

n = n1 × n2

for QD

2 4 6 8 10 12 14

n1

2

4

6

8

10

12

14

n2

0

0.05

0.1

0.15

0.2

0.25

A

Rank 2

(a)

2 4 6 8 10 12 14

n1

2

4

6

8

10

12

14

n2

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

A

Rank 4, NPPT

(b)

2 4 6 8 10 12 14

n1

2

4

6

8

10

12

14

n2

0

0.02

0.04

0.06

0.08

0.1

0.12

A

Rank 4, PPT



n =∑n2

i=1 ni1

0.04

0.06

0.08

0.1

0.12

0.14

5 10 15 20 25 30

– εnr

n2

Rank 2

Rank 3, NPPT

Rank 3, PPT

Rank 4, NPPT

Rank 4, PPT

n1 = 10

n = n1 × n2

for QD

2 4 6 8 10 12 14

n1

2

4

6

8

10

12

14

n2

0

0.05

0.1

0.15

0.2

0.25

A

Rank 2

(a)

2 4 6 8 10 12 14

n1

2

4

6

8

10

12

14

n2

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

A

Rank 4, NPPT

(b)

2 4 6 8 10 12 14

n1

2

4

6

8

10

12

14

n2

0

0.02

0.04

0.06

0.08

0.1

0.12

A

Rank 4, PPT



Triad

n = 3

-1-0.5

0 0.5

1 0

0.5 1

1.5 2

2.5 3

0

1

2

3

4

5

6

7

8

9

P(fθ, φ)

fθ

φ

P(fθ, φ)

fθ = cos θ

ρAB =14

[IA ⊗ IB +∑

α=x,y,z

cαασαA ⊗ σαB +∑

α=x,y,z

cAασ

αA ⊗ IB +

∑β=x,y,z

cBβ IA ⊗ σβB ]

Triad

n = 3

-1-0.5

0 0.5

1 0

0.5 1

1.5 2

2.5 3

0

1

2

3

4

5

6

7

8

9

P(fθ, φ)

fθ

φ

P(fθ, φ)

fθ = cos θ

ρAB =14

[IA ⊗ IB +∑

α=x,y,z


α=x,y,z

cAασ

αA ⊗ IB +

∑β=x,y,z

cBβ IA ⊗ σβB ]

Triad

n = 3

-1-0.5

0 0.5

1 0

0.5 1

1.5 2

2.5 3

0

1

2

3

4

5

6

7

8

9

P(fθ, φ)

fθ

φ

P(fθ, φ)

fθ = cos θ

ρAB =14

[IA ⊗ IB +∑

α=x,y,z


α=x,y,z

cAασ

αA ⊗ IB +

∑β=x,y,z

cBβ IA ⊗ σβB ]

Recapitulation

n points n =∑n2

i=1 ni1

PRA 92, 062301 (2015)

n = n1 × n2 n = 3

Only discord?

Recapitulation

n points n =∑n2

i=1 ni1

PRA 92, 062301 (2015)

n = n1 × n2 n = 3

Only discord?

How about other correlation measures?

Quantum work deficit: W(ρAB) = min{ΠA

k }

[S(∑

k pkρkAB)− S(ρAB)

]Horodecki et. al., PRA (2005)

for QD

2 4 6 8 10 12 14

n1

2

4

6

8

10

12

14

n2

0

0.05

0.1

0.15

0.2

0.25

A

Rank 2

(a)

for QWD

2 4 6 8 10 12 14

n1

2

4

6

8

10

12

14

n2

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

A

Rank 2

(d)

2 4 6 8 10 12 14

n1

2

4

6

8

10

12

14

n2

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

A

Rank 4, NPPT

(b)

2 4 6 8 10 12 14

n1

2

4

6

8

10

12

14

n2

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2

A

Rank 4, NPPT

(e)

2 4 6 8 10 12 14

n1

2

4

6

8

10

12

14

n2

0

0.02

0.04

0.06

0.08

0.1

0.12

A

Rank 4, PPT

2 4 6 8 10 12 14

n1

2

4

6

8

10

12

14

n2

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

A

Rank 4, PPT

Hierarchy in correlation measures

How about other correlation measures?Quantum work deficit: W(ρAB) = min

{ΠAk }

[S(∑



for QD

2 4 6 8 10 12 14

n1

2

4

6

8

10

12

14

n2

0

0.05

0.1

0.15

0.2

0.25

A

Rank 2

(a)

for QWD

2 4 6 8 10 12 14

n1

2

4

6

8

10

12

14

n2

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

A

Rank 2

(d)

2 4 6 8 10 12 14

n1

2

4

6

8

10

12

14

n2

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

A

Rank 4, NPPT

(b)

2 4 6 8 10 12 14

n1

2

4

6

8

10

12

14

n2

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2

A

Rank 4, NPPT

(e)

2 4 6 8 10 12 14

n1

2

4

6

8

10

12

14

n2

0

0.02

0.04

0.06

0.08

0.1

0.12

A

Rank 4, PPT

2 4 6 8 10 12 14

n1

2

4

6

8

10

12

14

n2

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

A

Rank 4, PPT


How about other correlation measures?Quantum work deficit: W(ρAB) = min

{ΠAk }

[S(∑



for QD

2 4 6 8 10 12 14

n1

2

4

6

8

10

12

14

n2

0

0.05

0.1

0.15

0.2

0.25

A

Rank 2

(a)

for QWD

2 4 6 8 10 12 14

n1

2

4

6

8

10

12

14

n2

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

A

Rank 2

(d)

2 4 6 8 10 12 14

n1

2

4

6

8

10

12

14

n2

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

A

Rank 4, NPPT

(b)

2 4 6 8 10 12 14

n1

2

4

6

8

10

12

14

n2

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2

A

Rank 4, NPPT

(e)

2 4 6 8 10 12 14

n1

2

4

6

8

10

12

14

n2

0

0.02

0.04

0.06

0.08

0.1

0.12

A

Rank 4, PPT

2 4 6 8 10 12 14

n1

2

4

6

8

10

12

14

n2

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

A

Rank 4, PPT


1. Correlations with low errorAnalytical expressions in terms of system parameters available

Example: Two-qubit XY model in inhomogeneous fieldH = J

{(1 + g)σx

1σx2 + (1− g)σy

1σy2

}+∑2

i=1 hiσzi

Thermal state: ρT = e−βH

Tr[e−βH ]; β = 1/kBT

(a)

"data.dat" u 1:2:4

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2h1/J

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

h2/J

0.1

0.14

0.18

0.22

0.26

(b)

"data.dat" u 1:2:6

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2h1/J

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

h2/J

0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55

For QD: Exact when SE→ Triad

For QWD

SE→ Triadεmax = 6.26× 10−2

(h1/J, h2/J) = (±1.45,±0.55)

SE→ Type 3εmax ∼ 10−6 for n1 = 8, n2 = 1

PRA 92, 062301 (2015)

1. Correlations with low errorAnalytical expressions in terms of system parameters available

Example: Two-qubit XY model in inhomogeneous fieldH = J

{(1 + g)σx

1σx2 + (1− g)σy

1σy2

}+∑2

i=1 hiσzi

Thermal state: ρT = e−βH

Tr[e−βH ]; β = 1/kBT

(a)

"data.dat" u 1:2:4

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2h1/J

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

h2/J

0.1

0.14

0.18

0.22

0.26

(b)

"data.dat" u 1:2:6

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2h1/J

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

h2/J

0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55

For QD: Exact when SE→ Triad

For QWD

SE→ Triadεmax = 6.26× 10−2

(h1/J, h2/J) = (±1.45,±0.55)

SE→ Type 3εmax ∼ 10−6 for n1 = 8, n2 = 1

PRA 92, 062301 (2015)





ρAB =14

[IA ⊗ IB +∑

α=x,y,z




Huang, NJP (2014)


2. Investigating decoherence“Freezing phenomena”

Canonical initial states under local quantum channels

Example: ρ̃AB under bit-flip channel

ρ̃AB =14

[IA ⊗ IB +∑

α=x,y,z

cαασαA ⊗ σαB + (cAx σ

xA ⊗ IB + cB

x IA ⊗ σxB)]

Necessary & sufficient initial conditions for freezing can be determined

PRA 91, 062119 (2015)

3. Higher dimensional systemsStates with positive partial transposition

Example:

%α =27|ψ〉〈ψ|+ α

7%+ +

5− α7

%−

%+ = (|01〉〈01| + |12〉〈12| + |20〉〈20|)/3%− = (|10〉〈10| + |21〉〈21| + |02〉〈02|)/3|ψ〉 = 1√

3

∑2i=0 |ii〉

0 ≤ α ≤ 5

Bound entangled: 1 < α ≤ 2, 3 < α ≤ 4Separable: 2 ≤ α ≤ 3Distillable: 0 < α ≤ 1, 4 < α ≤ 5

0.34

0.36

0.38

0.4

0.42

0.44

0.46

0 1 2 3 4 5

QD

α

DcDa

0×100

2×10-3

4×10-3

6×10-3

8×10-3

1×10-2

0 1 2 3 4 5

ε

α

PRA 92, 062301 (2015)

In a nutshell...

• Constrained quantum correlations

• Analytical formula with low error

• Applications in many-body physics, decoherence, higher dimensional systems

• Applicable to other scenarios involving optimization

Documents

Constrained Quantum CorrelationsConstrained …confqic/qipa15/talks/amit.pdfConstrained Quantum CorrelationsConstrained Quantum Correlations A user-friendly computational scheme Amit