EVOLUTION OF CONTINUOUS VARIABLE CORRELATIONS IN OPEN QUANTUM SYSTEMS

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1 Dedicated to Academician Aureliu Sandulescu s 80 th Anniversary EVOLUTION OF CONTINUOUS VARIABLE CORRELATIONS IN OPEN QUANTUM SYSTEMS AURELIAN ISAR 1,2 1 Department of Theoretical Physics, Horia Hulubei National Institute for Physics and Nuclear Engineering, 30 Reactorului, RO , POB-MG6, Bucharest-Magurele, Romania isar@theory.nipne.ro 2 Academy of Romanian Scientists, 54 Splaiul Independentei, Bucharest , Romania Received January 10, 2012 In the framework of the theory of open systems based on completely positive quantum dynamical semigroups, we give a description of the continuous variable quantum entanglement and quantum discord for a system consisting of two non-interacting modes embedded in a thermal environment. We describe the evolution of entanglement in terms of the covariance matrix for Gaussian input states. For all values of the temperature of the thermal reservoir, an initial separable unimodal squeezed state and a squeezed thermal state remain separable for all times. We study the time evolution of logarithmic negativity, which characterizes the degree of entanglement, and show that in the case of an entangled initial squeezed vacuum state and a squeezed thermal state, entanglement suppression (entanglement sudden death) takes place, for non-zero temperatures of the environment. Only for a zero temperature of the thermal bath the initial entangled squeezed state remains entangled for finite times. Independent of its type separable or entangled, the initial state evolves asymptotically to an equilibrium state which is always separable. We analyze the time evolution of the Gaussian quantum discord, which is a measure of all quantum correlations in the bipartite state, including entanglement, and show that quantum discord decays asymptotically in time under the effect of the thermal bath. Before the suppression of the entanglement, the qualitative evolution of quantum discord is very similar to that of the entanglement. We describe also the time evolution of the degree of classical correlations and of quantum mutual information, which measures the total correlations of the quantum system. Key words: Quantum entanglement, quantum discord, Gaussian states, open systems. PACS: Yz, Bg, Mn. This paper is written in the Honour of Professor Aureliu Emil Sandulescu, who had a crucial influence on my scientific formation. I am very pleased to express to him my gratitude and, on the occasion of his 80th Anniversary, to wish him a long life in good health and further success in his scientific activity. 1. INTRODUCTION The study of quantum correlations is a key issue in quantum information theory [1] and quantum entanglement represents the indispensable physical resource for (c) Rom. RJP Journ. 57(Nos. Phys., Vol. 1-2) 57, Nos. 1-2, 2012 P , Bucharest, 2012

2 2 Evolution of continuous variable correlations in open quantum systems 263 the description and performance of quantum information processing tasks, like quantum teleportation, cryptography, superdense coding and quantum computation [2]. However, entanglement does not describe all the non-classical properties of quantum correlations. Zurek [3, 4] defined the quantum discord as a measure of quantum correlations which includes entanglement of bipartite systems and it can also exist in separable states. Very recently, an operational interpretation was given to quantum discord in terms on consumption of entanglement in an extended quantum state merging protocol [5, 6]. The total amount of correlations contained in a quantum state is given by the quantum mutual information which is equal to the sum of the quantum discord and classical correlations [7]. In recent years there is an increasing interest in using non-classical entangled states of continuous variable systems in applications of quantum information processing, communication and computation [8]. 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. Due to the unavoidable interaction with the environment, any pure quantum state evolves into a mixed state and to describe realistically continuous variable quantum information processes it is necessary to take decoherence and dissipation into consideration. Decoherence and dynamics of quantum entanglement in continuous variable open systems have been intensively studied in the last years [9 24]. In this review we give a description, in the framework of the theory of open systems based on completely positive quantum dynamical semigroups, of the dynamics of the continuous variable quantum entanglement and quantum discord of a subsystem consisting of two uncoupled modes (harmonic oscillators) interacting with a common thermal environment. We are interested in discussing the correlation effect of the environment, therefore we assume that the two modes are independent, i.e. they do not interact directly. The initial state of the subsystem is taken of Gaussian form and the evolution under the quantum dynamical semigroup assures the preservation in time of the Gaussian form of the state. We have studied previously [25 32] the evolution of the entanglement of two identical harmonic oscillators interacting with a general environment, characterized by general diffusion and dissipation coefficients. We obtained that, depending on the values of these coefficients, the state keeps for all times its initial type: separable or entangled. In other cases, entanglement generation, entanglement sudden death or a periodic collapse and revival of entanglement take place. The paper is organized as follows. In Sec. 2 we write the Markovian master equation in the Heisenberg representation for two uncoupled harmonic oscillators interacting with a general environment and give the general solution of the evolution equation for the covariance matrix corresponding to a generic two-mode Gaussian state. We investigate in Sec. 3 the dynamics of entanglement for the considered

3 264 Aurelian Isar 3 subsystem and find that for all values of the temperature of the thermal reservoir, an initial separable unimodal squeezed state and a squeezed thermal state remain separable for all times. We analyze the time evolution of the logarithmic negativity, which characterizes the degree of entanglement of the quantum state, and show that in the case of an entangled initial squeezed vacuum state and a squeezed thermal state, entanglement suppression (entanglement sudden death) takes place, for nonzero temperatures of the environment. Only for a zero temperature of the thermal bath the initial entangled squeezed state remains entangled for all finite times, but in the limit of infinite time it evolves asymptotically to an equilibrium state which is always separable. In Sec. 4 we analyze the time evolution of the Gaussian quantum discord, which is a measure of all quantum correlations in the bipartite state, including entanglement, and show that quantum discord decays asymptotically in time under the effect of the thermal bath. This is contrast with the sudden death of entanglement. Before the suppression of the entanglement, the qualitative evolution of quantum discord is very similar to that of the entanglement. We describe also the time evolution of the degree of classical correlations and of quantum mutual information, which measures the total amount of correlations of the quantum system. A summary is given in Sec DYNAMICS OF TWO MODES INTERACTING WITH AN ENVIRONMENT We study the dynamics of a subsystem composed of two non-interacting modes in weak interaction with a thermal environment. In the axiomatic formalism based on completely positive quantum dynamical semigroups, the Markovian irreversible time evolution of an open system is described by the Lindblad-Kossakowski master equation [33]. We are interested in the set of Gaussian states, therefore we introduce such quantum dynamical semigroups that preserve this set during time evolution of the system [34]. A two-mode Gaussian state is entirely specified by its covariance matrix, which is a real, symmetric and positive matrix: σ xx (t) σ xpx (t) σ xy (t) σ xpy (t) ( σ(t) = σ xpx (t) σ pxpx (t) σ ypx (t) σ pxpy (t) A σ xy (t) σ ypx (t) σ yy (t) σ ypy (t) C T σ xpy (t) σ pxpy (t) σ ypy (t) σ pypy (t) ) C, (1) B where A, B and C are 2 2 Hermitian 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 modes. The equations of motion for the quantum correlations of the coordinates x,y and momenta p x,p y of the two modes

4 4 Evolution of continuous variable correlations in open quantum systems 265 are the following (T denotes the transposed matrix) [34]: where dσ(t) dt = Y σ(t) + σ(t)y T + 2D, (2) λ 1/m 0 0 Y = mω1 2 λ λ 1/m, (3) 0 0 mω2 2 λ D xx D xpx D xy D xpy D = D xpx D pxpx D ypx D pxpy D xy D ypx D yy D ypy, (4) D xpy D pxpy D ypy D pypy m is the mass, ω 1 and ω 2 are the frequencies of the non-resonant modes, and diffusion coefficients D xx,d xpx,... and dissipation constant λ are real quantities. The time-dependent solution of Eq. (2) is given by [34] σ(t) = M(t)[σ(0) σ( )]M T (t) + σ( ), (5) where the matrix M(t) = exp(y t) has to fulfill the condition lim t M(t) = 0. The values at infinity are obtained from the equation Y σ( ) + σ( )Y T = 2D. (6) To describe the dynamics of quantum correlations, we use two types of measures: logarithmic negativity for entanglement, and quantum discord. 3. EVOLUTION OF THE CONTINUOUS VARIABLE ENTANGLEMENT 3.1. TIME EVOLUTION OF ENTANGLEMENT AND LOGARITHMIC NEGATIVITY The characterization of the separability of continuous variable states using second-order moments of quadrature operators was given in Refs. [35,36]. For Gaussian states, whose statistical properties are fully characterized by just second-order moments, this criterion was proven to be necessary and sufficient: a Gaussian state is separable if and only if the partial transpose of its density matrix is non-negative (positive partial transpose (PPT) criterion). The 4 4 covariance matrix (1) (where all first moments can be set to zero by means of local unitary operations which do not affect the entanglement) contains four local symplectic invariants in form of the determinants of the block matrices A,B,C and covariance matrix σ. For Gaussian states, the measures of entanglement of bipartite systems are based on the invariants constructed from the elements of

5 266 Aurelian Isar 5 the covariance matrix [9, 13], which can be calculated from Eqs. (5), (6). In order to quantify the degree of entanglement of the states of the two modes it is suitable to use the logarithmic negativity. For a Gaussian density operator, the logarithmic negativity is completely defined by the symplectic spectrum of the partial transpose of the covariance matrix. It is given by E N = max{0, log 2 2 ν }, where ν is the smallest of the two symplectic eigenvalues of the partial transpose σ of the two-mode covariance matrix σ [13]: 2 ν 2 = 2 4detσ (7) and is the symplectic invariant (seralian), given by = deta + detb 2detC. In our model, the logarithmic negativity is calculated as [22, 25, 26, 37] where E N (t) = max{0, 1 2 log 2[4g(σ(t))]}, (8) g(σ(t)) = 1 (deta + detb) detc 2 ( [1 ] ) 2 1/2 (deta + detb) detc detσ(t). (9) 2 It determines the strength of entanglement for E N (t) > 0, and if E N (t) = 0, then the state is separable. We suppose that the asymptotic state of the considered open system is a Gibbs state corresponding to two independent quantum harmonic oscillators in thermal equilibrium at temperature T. Then the quantum diffusion coefficients have the following form (we put = 1) [33]: mω 1 D xx = D p xp x = λ mω 1 2 coth ω 1 2kT, mω 2D yy = D p yp y = λ mω 2 2 coth ω 2 2kT, D xpx = D ypy = D xy = D pxpy = D xpy = D ypx = 0. (10) In the following, we analyze the dependence of the logarithmic negativity E N (t) on time t and temperature T of the thermal bath, with the diffusion coefficients given by Eqs. (10). We consider three types of initial Gaussian states: 1) a separable initial Gaussian state, with the two modes initially prepared in their single-mode squeezed states (unimodal squeezed state) and with its initial co-

6 6 Evolution of continuous variable correlations in open quantum systems 267 variance matrix taken of the form cosh 2r sinh 2r 0 0 σ s (0) = 1 sinh 2r cosh 2r cosh 2r sinh 2r, (11) 0 0 sinh 2r cosh 2r where r denotes the squeezing parameter; 2) an entangled initial Gaussian state, taken of the form of a two-mode vacuum squeezed state, with the initial covariance matrix given by cosh 2r 0 sinh 2r 0 σ e (0) = 1 0 cosh2r 0 sinh2r 2 sinh 2r 0 cosh 2r 0 ; (12) 0 sinh2r 0 cosh2r 3) a two-mode squeezed thermal state, with the covariance matrix of the form [38] a 0 c 0 σ st (0) = 1 0 a 0 c 2 c 0 b 0, (13) 0 c 0 b with the matrix elements given by a = n 1 cosh 2 r + n 2 sinh 2 r cosh2r, b = n 1 sinh 2 r + n 2 cosh 2 r cosh2r, (14) c = 1 2 (n 1 + n 2 + 1)sinh2r, where n 1,n 2 are the average number of thermal photons associated with the two modes and r denotes the squeezing parameter. In the particular case n 1 = 0 and n 2 = 0, (11) 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 > r s [38], where cosh 2 r s = (n 1 + 1)(n 2 + 1). (15) n 1 + n The calculations with separable initial unimodal squeezed states (11) and squeezed thermal states (13) show that these states remain separable for all values of the temperature T and for all times. The evolution of entangled initial squeezed thermal states with the covariance matrix given by Eq. (11) is illustrated in Figure 1, where we represent the dependence of the logarithmic negativity E N (t) on time t and temperature T for the case of an initial symmetric Gaussian state, when a = b. We

7 268 Aurelian Isar 7 observe that for a non-zero temperature T, at certain finite moment of time, which depends on T, E N (t) becomes zero and therefore the state becomes separable. This is the so-called phenomenon of entanglement sudden death. It is in contrast to the quantum decoherence, during which the loss of quantum coherence is usually gradual [19, 39]. For T = 0, E N (t) remains strictly positive for finite times and tends asymptotically to 0 for t. Therefore, only for zero temperature of the thermal bath the initial entangled state remains entangled for finite times and this state tends asymptotically to a separable one for infinitely large time. One can also show that the dissipation favors the phenomenon of entanglement sudden death with increasing the dissipation parameter λ, the entanglement suppression happens earlier. The same qualitative behavior of the time evolution of entanglement was obtained previously [40 42] in the particular case n 1 = 0 and n 2 = 0 corresponding to an initial two-mode squeezed vacuum state. The dynamics of entanglement of the two oscillators depends strongly on the initial states and the coefficients describing the interaction of the system with the thermal environment (dissipation constant and temperature). As expected, the logarithmic negativity has a behavior similar to that one of the Simon function in what concerns the characteristics of the state of being separable or entangled [25, 26, 37, 39]. 4 E N T 10 t Fig. 1 Logarithmic negativity E N versus time t and temperature T for an entangled initial symmetric squeezed thermal state with squeezing parameter r = 2, n 1 = 1,n 2 = 1 and λ = 0.1,ω 1 = ω 2 = 1. We take m = = k = 1.

8 8 Evolution of continuous variable correlations in open quantum systems ASYMPTOTIC ENTANGLEMENT On general grounds, one expects that the effect of decoherence is dominant in the long-time regime, so that no entanglement is expected to be left at infinity. Indeed, using the diffusion coefficients given by Eqs. (10), we obtain from Eq. (6) the following elements of the asymptotic matrices A( ) and B( ): mω 1 σ xx ( ) = σ p xp x ( ) mω 1 = 1 2 coth ω 1 2kT, σ xp x ( ) = 0, mω 2 σ yy ( ) = σ p yp y ( ) mω 2 = 1 2 coth ω 2 2kT, σ yp y ( ) = 0 and of the entanglement matrix C( ): (16) σ xy ( ) = σ xpy ( ) = σ ypx ( ) = σ pxp y ( ) = 0. (17) Therefore the equilibrium asymptotic state is always separable in the case of two non-interacting modes immersed in a common thermal reservoir. In Refs. [25, 26, 32, 37, 39, 43] we described the dependence of the logarithmic negativity E N (t) on time and mixed diffusion coefficient for two harmonic oscillators interacting with a general environment. In the present case of a thermal bath, the asymptotic logarithmic negativity is given by (for ω 1 ω 2 ) E N ( ) = max{0, log 2 coth ω 2 }. (18) 2kT It depends only on temperature, and does not depend on the initial Gaussian state. E N ( ) = 0 both for T 0 and T = 0, and this confirms the previous statement that the asymptotic state is always separable. 4. EVOLUTION OF THE GAUSSIAN QUANTUM DISCORD The separability of quantum states has often been described as a property synonymous with the classicality. However, recent studies have shown that separable states, usually considered as being classically correlated, might also contain quantum correlations. Quantum discord was introduced [3, 4] as a measure of all quantum correlations in a bipartite state, including but not restricted to entanglement. Quantum discord has been defined as the difference between two quantum analogues of classically equivalent expression 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 state and therefore the correlations in such separable states with positive discord are an indicator of quantumness. States with zero discord represent essentially a classical probability distribution embedded in a quantum system.

9 270 Aurelian Isar 9 For an arbitrary bipartite state ρ 12, the total correlations are expressed by quantum mutual information [44] I(ρ 12 ) = S(ρ i ) S(ρ 12 ), (19) i=1,2 where ρ i represents the reduced density matrix of subsystem i and S(ρ) = Tr(ρlnρ) is the von Neumann entropy. A measure of bipartite classical correlations C(ρ 12 ) in the bipartite quantum state ρ 12 based on a complete set of local projectors {Π k 2 } on the subsystem 2 can be given by C(ρ 12 ) = S(ρ 1 ) inf {Π k 2 } {S(ρ 1 2)}, (20) where S(ρ 1 2 ) = k pk S(ρ k 1 ) is the conditional entropy of subsystem 1 and inf{s(ρ 1 2 )} represents the minimal value of the entropy with respect to a complete set of local measurements {Π k 2 }. Here, pk is the measurement probability for the kth local projector and ρ k 1 denotes the reduced state of subsystem 1 after the local measurements. Then the quantum discord is defined by D(ρ 12 ) = I(ρ 12 ) C(ρ 12 ). (21) Originally the quantum discord was defined and evaluated mainly for finite dimensional systems. Very recently [45, 46] 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 have been derived for bipartite thermal squeezed states [45] and for all two-mode Gaussian states [46]. The Gaussian quantum discord of a general two-mode Gaussian state ρ 12 can be defined as the quantum discord where the conditional entropy is restricted to generalized Gaussian positive operator valued measurements (POVM) on the mode 2, and in terms of symplectic invariants it is given by (the symmetry between the two modes 1 and 2 is broken) [46] where D = f( β) f(ν ) f(ν + ) + f( ε), (22) f(x) = x + 1 log x + 1 x 1 log x , (23) 2γ 2 + (β 1)(δ α) + 2 γ γ 2 + (β 1)(δ α) (β 1) 2 if (δ αβ) 2 (β + 1)γ 2 (α + δ) ε = αβ γ 2 + δ γ 4 + (δ αβ) 2 2γ 2 (24) (δ + αβ) 2β otherwise, α = 4detA, β = 4detB, γ = 4detC, δ = 16detσ, (25)

10 10 Evolution of continuous variable correlations in open quantum systems 271 and ν are the symplectic eigenvalues of the state, given by 2ν 2 = 2 4detσ, (26) where = det A + det B + 2 det C. Notice that Gaussian quantum discord only depends on detc, i.e., entangled (detc < 0) and separable states are treated on equal footing D T 10 t Fig. 2 Gaussian quantum discord D versus time t and temperature T for an entangled initial symmetric squeezed thermal state with squeezing parameter r = 2, n 1 = n 2 = 1 and λ = 0.1, ω 1 = ω 2 = 1. We take m = = k = 1. Explicit thresholds are shown for D = 0 and D = 1. The evolution of the Gaussian quantum discord D is illustrated in Figure 2, where we represent the dependence of D on time t and temperature T for an entangled initial symmetric Gaussian state, taken of the form of a two-mode squeezed thermal states (11), for such values of the parameters which satisfy for all times the first condition in formula (24). Gaussian discord asymptotically decreases in time, compared to the case of the logarithmic negativity, which has an evolution leading to a sudden suppression of the entanglement. For entangled initial states the Gaussian discord remains strictly positive in time and in the limit of infinite time it tends asymptotically to zero, corresponding to the thermal product (separable) state, with no correlations at all. The Gaussian discord has nonzero values for all finite times and this fact certifies the existence of nonclassical correlations in two-mode Gaussian states, either separable or entangled. One can easily show that for a separable initial Gaussian state with covariance matrix (11) the quantum discord is zero and it keeps this value during the whole time evolution of the state.

11 272 Aurelian Isar 11 From Figure 2 we notice that, in concordance with the general properties of the Gaussian quantum discord [46], the states can be either separable or entangled for D 1 and all the states above the threshold D = 1 are entangled. We also notice that the decay of quantum discord is stronger when the temperature T is increasing. It should be remarked that the decay of quantum discord is very similar to that of the entanglement before the time of the sudden death of entanglement. In the vicinity of a zero logarithmic negativity (E N = 0), the nonzero values of the discord can quantify the nonclassical correlations for separable mixed states and one considers that this fact could make possible some tasks in quantum computation [47]. The measure of classical correlations for a general two-mode Gaussian state ρ 12 can also be calculated and it is given by [46] C = f( α) f( ε), (27) while the expression of the quantum mutual information, which measures the total correlations, is given by I = f( α) + f( β) f(ν ) f(ν + ). (28) For an entangled initial Gaussian state, the degree of classical correlations C and, respectively, of the quantum mutual information I manifest a time evolution qualitatively similar to that one of the Gaussian discord: they have nonzero values for all finite times and in the limit of infinite time they tend asymptotically to zero, corresponding to the thermal product (separable) state, with no correlation at all [40]. In addition, the classical correlations and quantum mutual information decrease with increasing the temperature of the thermal bath. 5. SUMMARY In the framework of the theory of open systems based on completely positive quantum dynamical semigroups, we investigated the Markovian dynamics of quantum correlations for a subsystem composed of two non-interacting modes embedded in a thermal bath. We have presented and discussed the influence of the environment on the dynamics of quantum entanglement and quantum discord for different Gaussian initial states. We have described the time evolution of the logarithmic negativity in terms of the covariance matrix for the case when the asymptotic state of the considered open system is a Gibbs state corresponding to two independent quantum harmonic oscillators in thermal equilibrium. The dynamics of the quantum entanglement strongly depends on the initial states and the parameters characterizing the environment (dissipation coefficient and temperature). For all values of the temperature of the thermal reservoir, an initial separable Gaussian state remains separable for all times. In the case of an initial entangled squeezed vacuum state and squeezed

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