quantum discord and classical correlations of two bosonic modes in the two-reservoir model

Journal of Russian Laser Research, Volume 35, Number 1, January, 2014

QUANTUM DISCORD AND CLASSICAL CORRELATIONS

OF TWO BOSONIC MODES

IN THE TWO-RESERVOIR MODEL

Aurelian Isar

Horia Hulubei National Institute for Research and Development

in Physics and Nuclear Engineering

P.O. Box MG-6, Bucharest-Magurele, Romania∗E-mail: [email protected]

Abstract

Within the framework of the theory of open systems based on completely positive quantum dynamicalsemigroups, we give a description of the continuous variable quantum discord for a system consistingof two bosonic modes embedded in two independent thermal environments. We describe the evolutionof discord in terms of the covariance matrix for Gaussian input states. In the case of an entangledinitial squeezed thermal state, we analyze the evolution of the Gaussian quantum discord, which isa measure of all quantum correlations in the bipartite state, including entanglement, and show thatquantum discord decays asymptotically in time under the effect of the thermal reservoirs. We describealso the evolution of classical correlations and show that they also decay asymptotically in time.

Keywords: Gaussian states, open systems, Gaussian quantum discord, classical correlations.

1. Introduction

The realization of quantum information processing tasks depends on the generation and manipulation

of nonclassical states. In quantum information theory of continuous variable systems, Gaussian states,

in particular, two-mode Gaussian states, play a key role since they can be easily created and controlled

experimentally [1].

Quantum entanglement represents the indispensable physical resource for the description and per-

formance of quantum information processing tasks [2, 3]. However, entanglement does not describe all

the nonclassical properties of quantum correlations. Zurek [4, 5] defined the quantum discord as a mea-

sure of quantum correlations, which includes the entanglement of bipartite systems and can also exist

in separable states. Recently, an operational interpretation was given to quantum discord in terms of

consumption of the entanglement in an extended quantum state merging protocol [6, 7].

Quantum coherence and quantum correlations – entanglement and discord – of quantum systems are

inevitably influenced during their interaction with the external environment, and, in order to describe

realistically quantum information processes, it is necessary to take decoherence and dissipation into

consideration. In [8–11], we studied the dynamics of the Gaussian quantum discord of two bosonic modes

(two harmonic oscillators) coupled to a common thermal environment, employing the theory of open

systems based on completely positive quantum dynamical semigroups. Recently [12], we analyzed the

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Manuscript submitted by the author in English on November 19, 2013.

1071-2836/14/3501-0062 c©2014 Springer Science+Business Media New York

Volume 35, Number 1, January, 2014 Journal of Russian Laser Research

dynamics of the Gaussian quantum discord of two modes coupled to two independent thermal reservoirs

for initial two-mode squeezed vacuum states. In the present work, we extend this study to the general

case of initial two-mode squeezed thermal states. The initial state of the subsystem is assumed of a

Gaussian form, and the evolution under the quantum dynamical semigroup assures the preservation in

time of the Gaussian form of the state.

This paper is organized as follows.

In Sec. 2, we write the Markovian master equation for an open system interacting with a general

environment and the evolution equation for the covariance matrix. For this equation, we give its general

solution, i.e., we derive the variances and covariances of coordinates and momenta corresponding to

a one-mode Gaussian state. Using this solution, we obtain in Sec. 3 the evolution of the covariance

matrix for a Gaussian state of the two-mode bosonic system, where each mode interacting with its own

thermal reservoir. We analyze the time evolution of the Gaussian quantum discord, which is a measure

of all quantum correlations in the bipartite state, including entanglement. Then, using the asymptotic

covariance matrix, we show that quantum discord decays asymptotically in the limit of long times under

the effect of the thermal baths. We describe also the evolution of classical correlations and show that

they also decay asymptotically in time. We give our summary in Sec. 4.

2. Master Equation for Open Quantum Systems

We study the dynamics of a subsystem composed of two identical bosonic modes, each one in weak

interaction with its local thermal environment. In the axiomatic formalism based on completely positive

quantum dynamical semigroups, the irreversible time evolution of an open system is described by the

following Kossakowski–Lindblad quantum Markovian master equation for the density operator ρ(t) in

the Schrodinger representation († denotes the Hermitian conjugation) [13–15]

dρ(t)

dt= − i

�[H, ρ(t)] +

1

2�

∑j

(2Vjρ(t)V

†j − {ρ(t), V †

j Vj}+), (1)

where H denotes the Hamiltonian of the open system, and the operators Vj , V†j , defined on the Hilbert

space of H represent the interaction of the open system with the environment.

We are interested in the set of Gaussian states; therefore, we introduce such quantum dynamical

semigroups that preserve this set during the time evolution of the system. Consequently, H is taken

to be a polynomial of the second degree in the coordinates x, y and momenta px, py of the two bosonic

modes, and Vj , V†j are taken as polynomials of the first degree in these canonical observables.

We introduce the following 4×4 bimodal covariance matrix:

σ(t) =

⎛⎜⎜⎜⎜⎝

σxx(t) σxpx(t) σxy(t) σxpy(t)

σxpx(t) σpxpx(t) σypx(t) σpxpy(t)

σxy(t) σypx(t) σyy(t) σypy(t)

σxpy(t) σpxpy(t) σypy(t) σpypy(t)

⎞⎟⎟⎟⎟⎠ , (2)

with the correlations of operators A1 and A2 defined by using the density operator ρ of the initial state

of the quantum system as follows:

σA1A2(t) =1

2Tr[ρ(A1A2 +A2A1)(t)]− Tr[ρA1(t)]Tr[ρA2(t)]. (3)

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Journal of Russian Laser Research Volume 35, Number 1, January, 2014

A Gaussian channel is a map that converts the Gaussian states to the Gaussian states, and the

evolution given by Eq. (1) is such a channel. The evolution of the initial covariance matrix σ(0) of the

system, under the action of a general Gaussian channel, can be characterized by two matrices X(t) and

Y (t)

σ(t) = X(t)σ(0)XT(t) + Y (t), (4)

where Y (t) is a positive operator [16], with T denoting the transposed matrix. These two matrices com-

pletely characterize the action of environment, and Eq. (4) guarantees that σ(t) is a physical covariance

matrix for all finite times t.

In the case of one bosonic mode (harmonic oscillator) with the general Hamiltonian

H =1

2mp2x +

mω2

2x2 +

μ

2(xpx + pxx), (5)

where the parameter μ determines the damping ratio, we obtain from Eq. (1) the following system of

equations for the elements of the covariance matrix:

s(t) =

(σxx(t) σxpx(t)

σxpx(t) σpxpx(t)

), (6)

representing the quantum correlations of the canonical observables of a single-mode system [15,17]

ds(t)

dt= Z1s(t) + s(t)ZT

1 + 2D1, (7)

where from now we assume, for simplicity, m = ω = � = 1,

Z1 =

(−(λ1 − μ) 1

−1 −(λ1 + μ)

), D1 =

(Dxx Dxpx

Dxpx Dpxpx

), (8)

with Dxx, Dxpx , Dpxpx the diffusion coefficients, and λ1 is the dissipation constant. The solution of Eq. (7)

is given by [17]

s(t) = X1(t)[s(0)− s(∞)]XT1 (t) + s(∞), (9)

where the matrix X1(t) = exp(Z1t) has to fulfill the condition limt→∞X1(t) = 0.

For such a limit to exist, Z1 must have only eigenvalues with negative real parts. In the underdamped

case ω > μ and Ω2 ≡ ω2 − μ2, Eq. (9) takes the form

s(t) = X1(t)s(0)XT1 (t) + Y1(t), (10)

where

X1(t) = e−λ1t

(cosΩt+ (μ/Ω) sinΩt (1/Ω) sinΩt

−(1/Ω) sinΩt cosΩt− (μ/Ω) sinΩt

)(11)

and

Y1(t) = −X1(t)s(∞)XT1 (t) + s(∞). (12)

In the case where the asymptotic state is the Gibbs state corresponding to a harmonic oscillator in

the thermal equilibrium at temperature T1, the values at infinity are obtained from the equation

Z1s(∞) + s(∞)ZT1 = −2D1, (13)

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where the matrix of diffusion coefficients becomes

D1 =

([(λ1 − μ)/2] coth(2kT1)

−1 0

0 [(λ1 + μ)/2] coth(2kT1)−1

). (14)

3. Dynamics of the Gaussian Quantum Discord

A two-mode Gaussian state is entirely specified by its covariance matrix (2), which is a real, symmetric,

and positive matrix with the following block structure:

σ(t) =

(A C

CT B

), (15)

where A, B, and C are 2×2 Hermitian matrices. The matrices A and B denote the symmetric covariance

matrices for the individual reduced one-mode states, while the matrix C contains the cross correlations

between the modes. The covariance matrix (15) (where all first moments have been set to zero by

means of local unitary operations that do not affect the Gaussian discord) contains four local symplectic

invariants in the form of the determinants of the block matrices A,B,C and covariance matrix σ.

Consider a system of two identical bosonic modes coupled to two local thermal baths. If the initial

two-mode 4×4 covariance matrix is σ(0), its subsequent evolution is given by

σ(t) =(X1(t)⊕X2(t)

)σ(0)

(X1(t)⊕X2(t)

)T+

(Y1(t)⊕ Y2(t)

), (16)

where X1,2(t) and Y1,2(t) are given by Eqs. (11) and (12) and the similar ones, corresponding to each

bosonic mode immersed in its own thermal reservoir characterized by temperatures T1, respectively T2,

and dissipation constants λ1 and, respectively, λ2.

Quantum discord was introduced [4, 5] as a measure of all quantum correlations in a bipartite state,

including – but not restricted to – entanglement. Quantum discord was defined as the difference between

two quantum analogs of classically equivalent expressions of the mutual information, which is a measure

of total correlations in a quantum state. For pure entangled states, quantum discord coincides with the

entropy of entanglement. Quantum discord can be different from zero also for some mixed separable

states, and therefore the correlations in such separable states with positive discord are an indicator of the

quantumness. States with zero discord represent essentially a classical probability distribution embedded

in a quantum system.

For an arbitrary bipartite state ρ12, the total correlations are expressed by quantum mutual informa-

tion [18]

I(ρ12) =∑i=1,2

S(ρi)− S(ρ12), (17)

where ρi represents the reduced density matrix of subsystem i, and S(ρ) = −Tr(ρ ln ρ) is the von Neumann

entropy. Henderson and Vedral [19] proposed a measure of bipartite classical correlations C(ρ12) based

on a complete set of local projectors {Πk2} on the subsystem 2 — the classical correlation in the bipartite

quantum state ρ12 can be given by

C(ρ12) = S(ρ1)− inf{Πk2}{S(ρ1|2)}, (18)

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where S(ρ1|2) =∑

k pkS(ρk1) is the conditional entropy of subsystem 1, and inf{S(ρ1|2)} represents the

minimum value of the entropy with respect to a complete set of local measurements {Πk2}. Here, pk is

the measurement probability for the kth local projector, and ρk1 denotes the reduced state of subsystem

1 after the local measurements. Then the quantum discord is defined by

D(ρ12) = I(ρ12)− C(ρ12). (19)

Originally, the quantum discord was defined and evaluated mainly for finite dimensional systems.

Recently [20,21], the notion of discord has been extended to the domain of continuous variable systems,

in particular, to the analysis of bipartite systems described by two-mode Gaussian states. Closed formulas

were derived for bipartite thermal squeezed states [20] and all two-mode Gaussian states [21].

The Gaussian quantum discord of a general two-mode Gaussian state ρ12 can be defined as the

quantum discord where the conditional entropy is restricted to generalized Gaussian positive operator-

valued measurements (POVM) on mode 2, and in terms of symplectic invariants it is given by (the

symmetry between modes 1 and 2 is broken) [21]

D = f(√

β)− f(ν−)− f(ν+) + f(√ε), (20)

where

f(x) =x+ 1

2log

x+ 1

2− x− 1

2log

x− 1

2, (21)

ε =

⎧⎪⎪⎪⎨⎪⎪⎪⎩

2γ2 + (β − 1)(δ − α) + 2|γ|√γ2 + (β − 1)(δ − α)

(β − 1)2, if (δ − αβ)2 ≤ (β + 1)γ2(α+ δ),

αβ − γ2 + δ −√γ4 + (δ − αβ)2 − 2γ2(δ + αβ)

2β, otherwise,

(22)

α = 4 detA, β = 4 detB, γ = 4 detC, δ = 16 detσ, (23)

and ν∓ are the symplectic eigenvalues of the state given by

2ν2∓ = Δ∓√Δ2 − 4δ, (24)

where Δ = α+β+2γ. Notice that the Gaussian quantum discord depends only on | detC|, i.e., entangled(detC < 0) and separable states are treated on equal footing.

The evolution of the Gaussian quantum discord D (20) is illustrated in Fig. 1, where we represent

the dependence of the Gaussian discord D on time t, damping parameter μ, temperatures T1 and T2, and

dissipation constants λ1 and λ2 of the two thermal reservoirs, when the diffusion coefficients are given

by Eq. (14) and a similar one for the second reservoir.

We assume that the initial Gaussian state is a two-mode squeezed thermal state, with the covariance

matrix of the form [22]

σst(0) =

⎛⎜⎜⎜⎜⎝a 0 c 0

0 a 0 −c

c 0 b 0

0 −c 0 b

⎞⎟⎟⎟⎟⎠ , (25)

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Fig. 1. Gaussian quantum discord D versus time t and temperature T2 (via coth 1/2kT2) for an initial squeezedthermal state with r = 0.8, λ1 = 0.3, λ2 = 0.2, μ = 0.1, n1 = 1, n2 = 3, coth 1/2kT1 = 1.1 (left) and r = 1, λ1 =0.03, λ2 = 0.05, μ = 0.02, n1 = 2, n2 = 4, coth 1/2kT1 = 2 (right). We assume m = ω = � = 1.

with the matrix elements given by

a = n1 cosh2 r + n2 sinh

2 r +1

2cosh 2r, b = n1 sinh

2 r + n2 cosh2 r +

1

2cosh 2r,

(26)

c =1

2(n1 + n2 + 1) sinh 2r,

where n1, n2 are the average number of thermal photons associated with the two modes, and r denotes

the squeezing parameter. In the particular case n1 = 0 and n2 = 0, (25) becomes the covariance matrix

of the two-mode squeezed vacuum state. A two-mode squeezed thermal state is entangled when the

squeezing parameter r satisfies the inequality r > rs [22], where

cosh2 rs =(n1 + 1)(n2 + 1)

n1 + n2 + 1. (27)

The Gaussian discord has nonzero values for all finite times, and this fact certifies the existence of

nonclassical correlations in two-mode Gaussian states. The dynamics of discord of the two modes strongly

depends on the parameters characterizing the system and the coefficients describing the interaction of

the system with both reservoirs: quantum discord increases with increase in the squeezing parameter

and decreases with increase in the damping parameter, temperatures, and dissipation constants. The

evolution of D manifests a monotonically decreasing behavior. The Gaussian discord asymptotically

decreases to zero for large times. This behavior has to be compared with that of the entanglement,

which was recently shown [23] to manifest, in the same system and for the same initial Gaussian states,

a suppression at some finite moment of time.

A similar behavior of the discord was also obtained in the case of two bosonic modes interacting with

a common thermal bath, and for this system the logarithmic negativity has again the evolution leading

to a sudden death of the entanglement [24–27].

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Fig. 2. Classical correlations C versus time t and temperature T2 (via coth 1/2kT2) for an initial squeezed thermalstate with with r = 0.8, λ1 = 0.3, λ2 = 0.2, μ = 0.1, n1 = 1, n2 = 3, coth 1/2kT1 = 1.1 (left) and r = 1,λ1 = 0.03, λ2 = 0.05, μ = 0.02, n1 = 2, n2 = 4, coth 1/2kT1 = 2 (right). We assume m = ω = � = 1.

3.1. Classical Correlations

The measure of classical correlations for a general two-mode Gaussian state ρ12 can also be calculated;

it reads [21]

C = f(√α)− f(

√ε), (28)

while the expression of the quantum mutual information, which measures the total correlations, is

I = f(√α) + f(

√β)− f(ν−)− f(ν+). (29)

In Fig. 2, we illustrate the evolution of classical correlations C as a function of time t and temperature

T2 for an initial Gaussian state, taken in the form of a two-mode squeezed thermal state (25). This

quantity manifests a qualitative behavior similar to that of the Gaussian discord — it has nonzero values

for all finite times and, in the limit of infinite time it tends asymptotically to zero, corresponding to

the thermal product (separable) state, with no correlations at all. One can also see that the classical

correlations decrease with increase in the temperature of the thermal bath. One can also remark that,

in the considered cases, the values of classical correlations are larger than those of quantum correlations,

represented by the Gaussian quantum discord.

3.2. Asymptotic State

We assume that the asymptotic state of the considered open system is the Gibbs state corresponding

to two independent quantum harmonic oscillators in thermal equilibrium with their own reservoirs at

temperatures T1 and T2. On general grounds, one expects that the effect of decoherence is dominant in

the long-time regime, so that no quantum correlations are expected to be left at infinity. Indeed, using

the diffusion coefficients given by Eq. (14) and a similar one for the second thermal bath, we obtain from

68


Eq. (13) the following elements of the asymptotic matrices A(∞) and B(∞) :

σxx(∞) = σpxpx(∞) = 2−1 coth(2kT1)−1, σxpx(∞) = 0,

(30)σyy(∞) = σpypy(∞) = 2−1 coth(2kT2)

−1, σypy(∞) = 0,

while all the elements of the entanglement matrix C(∞) are zero. Then the discord expression (20)

becomes zero in the limit of large times and, correspondingly, the equilibrium asymptotic state has

always zero discord in the case of two identical bosonic modes immersed in two independent thermal

reservoirs, corresponding to the thermal product (separable) state, with no correlation at all, as in the

case of the same system immersed in a common thermal bath.

4. Summary

Within the framework of the theory of open quantum systems based on completely positive quantum

dynamical semigroups, we investigated the Markovian dynamics of the Gaussian quantum discord for

a subsystem composed of two bosonic modes, each one embedded in its own thermal bath. We have

presented and discussed the influence of the reservoirs on the dynamics of quantum discord in terms

of the covariance matrix for squeezed thermal initial states. We assumed that the asymptotic state of

the considered open system is the Gibbs state corresponding to two independent quantum harmonic

oscillators, each one in thermal equilibrium with its thermal bath.

The Gaussian discord has nonzero values for all finite times, and its dynamics strongly depends on the

parameters characterizing the system (squeezing parameter and damping parameter) and the coefficients

describing the interaction of the system with both reservoirs (temperatures and dissipation constants).

The values of the Gaussian discord asymptotically decrease to zero for large times. We described also

the time evolution of classical correlations. Presently there is a large debate relative to the physical

interpretation of quantum correlations – quantum entanglement and quantum discord. The present

results, in particular, the existence of quantum discord and the possibility of maintaining it in thermal

environments for long times, might be useful in controlling quantum correlations in open systems and

also for applications in the field of quantum information processing and communication.

Acknowledgments

The author acknowledges financial support of the Romanian Ministry of Education and Research

under Projects Nos. CNCS-UEFISCDI PN-II-ID-PCE-2011-3-0083 and PN 09 37 01 02/2010.

References

1. S. L. Braunstein and P. van Loock, Rev. Mod. Phys., 77, 513 (2005).

2. M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information, Cambridge

University Press (2000).

3. R. Horodecki, P. Horodecki, M. Horodecki, and K. Horodecki, Rev. Mod. Phys., 81, 865 (2009).

4. W. H. Zurek, Ann. Phys. (Leipzig), 9, 853 (2000).

5. H. Ollivier and W. H. Zurek, Phys. Rev. Lett., 88, 017901 (2001).

69


6. V. Madhok and A. Datta, Phys. Rev. A, 83, 032323 (2011).

7. D. Cavalcanti, L. Aolita, S. Boixo, et al., Phys. Rev. A, 83, 032324 (2011).

8. A. Isar, Open Sys. Inform. Dyn., 18, 175 (2011).

9. A. Isar, Phys. Scr., T147, 014015 (2012).

10. A. Isar, Phys. Scr. T153, 014035 (2013).

11. A. Isar, Phys. Scr., 87, 038108 (2013).

12. A. Isar, Open Sys. Inform. Dynam., 20, 1340003 (2013).

13. V. Gorini, A. Kossakowski, and E. C. G. Sudarshan, J. Math. Phys., 17, 821 (1976).

14. G. Lindblad, Commun. Math. Phys., 48, 119 (1976).

15. A. Isar, A. Sandulescu, H. Scutaru, et al., Int. J. Mod. Phys. E, 3, 635 (1994).

16. T. Heinosaari, A. S. Holevo, and M. M. Wolf, Quantum Inform. Comput., 10, 0619 (2010).

17. A. Sandulescu and H. Scutaru, Ann. Phys. (N.Y.), 173, 277 (1987).

18. B. Groisman, S. Popescu, and A. Winter, Phys. Rev. A, 72, 032317 (2005).

19. L. Henderson and V. Vedral, J. Phys. A: Math. Gen., 34, 6899 (2001).

20. P. Giorda and M. G. A. Paris, Phys. Rev. Lett., 105, 020503 (2010).

21. G. Adesso and A. Datta, Phys. Rev. Lett., 105, 030501 (2010).

22. O. Marian, T. A. Marian, and H. Scutaru, Phys. Rev. A, 68, 062309 (2003).

23. A. Isar, Rom. J. Phys., 65, 711 (2013).

24. A. Isar, J. Russ. Laser Res., 28, 439 (2007).

25. A. Isar, Int. J. Quantum Inform., 6, 689 (2008).

26. A. Isar, J. Russ. Laser Res., 31, 182 (2010).

27. A. Isar, Phys. Scr., T143, 014012 (2011).

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quantum discord and classical correlations of two bosonic modes in the two-reservoir model

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