QUANTUM ENTANGLEMENT OF TWO BOSONIC MODESIN TWO-RESERVOIR MODEL∗

AURELIAN ISAR

Department of Theoretical Physics“Horia Hulubei” National Institute for Physics and Nuclear EngineeringReactorului 30, RO-077125, POB-MG6, Magurele-Bucharest, Romania

E-mail: isar@theory.nipne.ro

Received February 20, 2013

In the framework of the theory of open systems based on completely positivequantum dynamical semigroups, we give a description of the continuous variable en-tanglement for a system consisting of two non-interacting bosonic modes embeddedin two independent thermal environments. By using the Peres-Simon necessary andsufficient criterion for separability of two-mode Gaussian states, we describe the evo-lution of entanglement in terms of the covariance matrix for Gaussian input states. Forall temperatures of the thermal reservoirs, an initial separable non-squeezed Gaussianstate remains separable for all times. In the case of an entangled initial squeezed va-cuum state, entanglement suppression (entanglement sudden death) takes place, for alltemperatures of the thermal baths. For definite values of temperatures and dissipa-tion constants, one can observe temporary revivals of the entanglement, but the systemevolves asymptotically to an equilibrium state which is always separable.

Key words: Quantum entanglement, open systems, Gaussian states.

PACS: 03.65.Yz, 03.67.Bg, 03.67.Mn.

1. INTRODUCTION

In recent years there is an increasing interest in using continuous variable sys-tems in applications of quantum information processing, communication and compu-tation [1]. The realization of quantum information processing tasks depends on thegeneration and manipulation of nonclassical states. In the special case of Gaussianstates there exist necessary and sufficient criteria of entanglement [2, 3] and quanti-tative entanglement measures [4, 5]. In quantum information theory of continuousvariable systems, Gaussian states, in particular two-mode Gaussian states, play a keyrole since they can be easily created and controlled experimentally. Implementa-tion of quantum communication and computation encounters the difficulty that any

∗Paper presented at “The 8th Workshop on Quantum Field Theory and Hamiltonian Systems”,September 19-22, 2012, Craiova, Romania.

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600 Aurelian Isar 2

realistic quantum system cannot be isolated and it always has to interact with its en-vironment. Quantum coherence and entanglement of quantum systems are inevitablyinfluenced during their interaction with the external environment and in order to de-scribe realistically quantum information processes it is necessary to take decoherenceand dissipation into consideration. Decoherence and dynamics of quantum entangle-ment in continuous variable open systems have bee intensively studied in the lastyears [6-18]. The Markovian time evolution of quantum correlations of entangledtwo-mode continuous variable states has been examined in single-reservoir [19] andtwo-reservoir models [20, 21], representing noisy correlated or uncorrelated Marko-vian quantum channels.

In the previous papers [22-26] we studied, in the framework of the theory ofopen systems based on completely positive quantum dynamical semigroups, the dy-namics of the entanglement of two bosonic modes (two identical harmonic oscilla-tors) coupled to a common thermal environment. In the present work we study thedynamics of the entanglement of two modes coupled to two independent thermalreservoirs. We are interested in discussing the correlation effect of the environment,therefore we assume that the two modes are uncoupled, i.e. they do not interactdirectly. The initial state of the subsystem is taken of Gaussian form and the evolu-tion under the quantum dynamical semigroup assures the preservation in time of theGaussian form of the state.

The paper is organized as follows. In Sec. 2 we write the Markovian mas-ter equation in the Heisenberg representation for an open system interacting with ageneral environment and the evolution equation for the covariance matrix. For thisequation we give its general solution, i.e. we derive the variances and covariances ofcoordinates and momenta corresponding to an one-mode Gaussian state. Using thissolution, we obtain in Sec. 3 the evolution of the covariance matrix for a Gaussianstate of the two-mode bosonic system, each mode interacting with its own thermalreservoir. By using the Peres-Simon necessary and sufficient condition for separabi-lity of two-mode Gaussian states [2,27], we investigate the dynamics of entanglementfor the considered open system. In particular, using the asymptotic covariance ma-trix, we determine the behaviour of the entanglement in the limit of long times. Weshow that for all values of the temperature of the thermal reservoirs, an initial sepa-rable non-squeezed Gaussian state remains separable for all times. In the case of anentangled initial squeezed vacuum state, entanglement suppression takes place, forall temperatures of the reservoirs, including zero temperatures. For definite valuesof temperatures and dissipation constants, one can observe temporary revivals of theentanglement, but the system evolves asymptotically to an equilibrium state which isalways separable. A summary is given in Sec. 4.

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2. MASTER EQUATION FOR OPEN QUANTUM SYSTEMS

We study the dynamics of a subsystem composed of two identical non-interactingbosonic modes, each one in weak interaction with its local thermal environment.In the axiomatic formalism based on completely positive quantum dynamical semi-groups, the irreversible time evolution of an open system is described by the follow-ing general quantum Markovian master equation for the density operator ρ(t) in theSchrodinger representation († denotes Hermitian conjugation) [28, 29]:

dρ(t)

dt=− i

~[H,ρ(t)]+

1

2~∑j

(2Vjρ(t)V†j −ρ(t),V †

j Vj+). (1)

Here, H denotes the Hamiltonian of the open system and the operators Vj ,V†j , de-

fined on the Hilbert space of H, represent the interaction of the open system with theenvironment.

We are interested in the set of Gaussian states, therefore we introduce suchquantum dynamical semigroups that preserve this set during time evolution of thesystem. Consequently H is taken to be a polynomial of second degree in the co-ordinates x,y and momenta px,py of the two bosonic modes and Vj ,V

†j are taken

polynomials of 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 ρof the initial state of the quantum system, as follows:

σA1A2(t) =1

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

A Gaussian channel is a map that takes Gaussian states to Gaussian states,and the evolution given by Eq. (1) is such a channel. The evolution of the initialcovariance 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 [30]. These two matrices completely characterizethe action of environment. Eq. (4) guarantees that σ(t) is a physical covariancematrix for all finite times t.

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

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602 Aurelian Isar 4

tonian

H =1

2mp2x+

mω2

2x2+

µ

2(pxx+xpx), (5)

we obtain from Eq. (1) the following system of equations for the elements of thecovariance matrix

s(t) =

(σxx(t) σxpx(t)σxpx(t) σpxpx(t)

), (6)

representing the quantum correlations of the canonical observables of a single-modesystem [29, 31]:

ds(t)

dt= Zs(t)+s(t)ZT+2D, (7)

where (we take from now on, for simplicity, m= ω = ~= 1)

Z =

(−(λ1−µ) 1

−1 −(λ1+µ)

), D =

(Dxx Dxpx

Dxpx Dpxpx

), (8)

Dxx,Dxpx ,Dpxpx are the diffusion coefficients and λ1 the dissipation constant. Thetime-dependent solution of Eq. (7) is given by [31]

s(t) =X(t)[s(0)−s(∞)]XT(t)+s(∞), (9)

where the matrix X(t) = exp(Zt) has to fulfill the condition limt→∞X(t) = 0. Inorder that this limit exists, Z must only have eigenvalues with negative real parts. Inthe underdamped case ω > µ, Ω2 ≡ ω2−µ2, it takes the form

s(t) =X(t)s(0)XT(t)+Y (t), (10)

where

X(t) = e−λ1t

(cosΩt+ µ

Ω sinΩt 1Ω sinΩt

− 1Ω sinΩt cosΩt− µ

Ω sinΩt

)(11)

and

Y (t) =−X(t)s(∞)XT(t)+s(∞). (12)

In the case when the asymptotic state is a Gibbs state corresponding to a har-monic oscillator in thermal equilibrium at temperature T1, the values at infinity areobtained from the equation

Zs(∞)+s(∞)ZT =−2D, (13)

where the matrix of diffusion coefficients becomes

D =

(λ1−µ

2 coth 12kT1

0

0 λ1+µ2 coth 1

2kT1

). (14)

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3. DYNAMICS OF TWO-MODE CONTINUOUS VARIABLE ENTANGLEMENT

In the following we remind the quantum separability criteria for a special classof states of continuous variable systems – the two-mode Gaussian states. A Gaussianstate has a Gaussian Wigner function in phase space and it is completely characte-rized by its first and second moments of canonical variables.

A well-known sufficient condition for inseparability is the so-called Peres-Horodecki criterion [27,32], which is based on the observation that the non-completelypositive nature of the partial transposition operation of the density matrix for a bi-partite system (this means transposition with respect to degrees of freedom of onesubsystem only) may turn an inseparable state into a non-physical state. The cha-racterization of the separability of continuous variable states using second-order mo-ments of quadrature operators was given in Refs. [2, 3]. For Gaussian states, whosestatistical properties are fully characterized by just second-order moments, this crite-rion was proven to be necessary and sufficient: a Gaussian continuous variable stateis separable if and only if the partial transpose of its density matrix is non-negative[positive partial transpose (PPT) criterion].

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 CCT B

), (15)

where A, B and C are 2× 2 Hermitian matrices. A and B denote the symmetriccovariance matrices for the individual reduced one-mode states, while the matrix Ccontains the cross-correlations between modes.

Consider a system of two identical bosonic modes coupled to two local thermalbaths. If the initial two-mode 4× 4 covariance matrix is σ(0), then its subsequentevolution 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), (12) and similar ones, correspond-ing to each bosonic mode immersed in its own thermal reservoir.

The covariance matrix (15) (where all first moments have been set to zero bymeans of local unitary operations which do not affect the entanglement) containsfour local symplectic invariants in form of the determinants of the block matricesA,B,C and covariance matrix σ. Based on the above invariants Simon [2] deriveda PPT criterion for bipartite Gaussian continuous variable states: the necessary and

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604 Aurelian Isar 6

sufficient criterion for separability is S(t)≥ 0, where

S(t)≡ detAdetB+(1

4−|detC|)2

−Tr[AJCJBJCTJ ]− 1

4(detA+detB)

(17)

and J is the 2×2 symplectic matrix

J =

(0 1−1 0

). (18)

This is also a necessary separability criterion for non-Gaussian states. According tothe previous results, the cross-correlations between modes can have non-zero values.In this case the Gaussian states with detC ≥ 0 are separable states, but for detC < 0it may be possible that the states are entangled.

In order to describe the dynamics of entanglement, we use the PPT crite-rion [2, 27] according to which a state is entangled if and only if the operation ofpartial transposition does not preserve its positivity. In the following, we analysethe dependence of the Simon function S(t) on time t and temperatures T1,T2 of thetwo thermal reservoirs, when the diffusion coefficients are given by Eq. (14) and thesimilar one for the second reservoir. We consider two types of the initial Gaussianstate: 1) separable and 2) entangled.

1) If the initial state is separable and non-squeezed, then the Simon functionbecomes strictly positive after the initial moment of time (S(0) = 0), so that theinitial separable state remains separable for all temperatures and all times.

2) The evolution of an entangled initial state is illustrated in Figs. 1,2, wherewe represent the dependence of the function S(t) on time t and temperature T2 foran entangled initial squeezed vacuum state of the form

σ(0) =

cosh2r 0 sinh2r 0

0 cosh2r 0 −sinh2rsinh2r 0 cosh2r 0

0 −sinh2r 0 cosh2r

, (19)

where r denotes the squeezing parameter. We observe that for a given temperatureT1 and all temperatures T2, at certain finite moment of time, which depends on T2,S(t) becomes zero and therefore the state becomes separable. This is the so-calledphenomenon of entanglement sudden death. This behaviour is in contrast to thatone of the quantum decoherence, during which the loss of quantum coherence isusually gradual [14, 33]. For definite values of temperatures T1,T2 and dissipationconstants λ1,λ2, one can observe temporary revivals of the entanglement, but thesystem evolves asymptotically to an equilibrium state which is always separable.

The dynamics of entanglement of the two modes strongly depends on the initial

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0

5

10

t

1

2

3

4

CothH12kT2L

0

5

10

15

20

S

Fig. 1 – Separability function S versus time t and environment temperature T2 (via coth 12kT2

) forλ= 0.1, λ= 0.2, µ= 0.9, coth 1

2kT1= 2, and entangled initial squeezed vacuum state with r = 0.5.

We take m= ω = ~= 1.

0

5

10

15

t 2

4

6

8

CothH12kT2L

-2000

-1000

0

1000

2000

S

Fig. 2 – Same as in Fig. 1, for λ= 0.01, µ= 0.2, coth 12kT1

= 3 and r = 0.5.

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states and the coefficients describing the interaction of the system with both reser-voirs (temperatures and dissipation constants).

3.1. ASYMPTOTIC STATE

On general grounds, one expects that the effects of decoherence, counteractingentanglement production, is dominant in the long-time regime, so that no quantumcorrelations (entanglement) is expected to be left at infinity. Indeed, using the diffu-sion coefficients given by Eq. (14) and the similar one for the second thermal bath,we obtain from Eq. (13) the following elements of the asymptotic matrices A(∞)and B(∞) :

σxx(∞) = σpxpx(∞) =1

2coth

1

2kT1, σxpx(∞) = 0,

σyy(∞) = σpypy(∞) =1

2coth

1

2kT2, σypy(∞) = 0,

(20)

while all the elements of the entanglement matrix C(∞) are zero:

σxy(∞) = σxpy(∞) = σypx(∞) = σpxpy(∞) = 0. (21)

Then the Simon expression (17) takes the following form in the limit of large times:

S(∞) =1

16(coth2

1

2kT1−1)(coth2

1

2kT2−1), (22)

and, correspondingly, the equilibrium asymptotic state is always separable in thecase of two identical non-interacting bosonic modes immersed in two independentthermal reservoirs.

The logarithmic negativity, which characterizes the degree of entanglement ofthe quantum state, is calculated as

EN (t) =−1

2log2[4f(σ(t))], (23)

where

f(σ(t)) =1

2(detA+detB)−detC

−

([1

2(detA+detB)−detC

]2−detσ(t)

)1/2

.

(24)

It determines the strength of entanglement for EN (t)> 0, and if EN (t)≤ 0, then thestate is separable.

In Refs. [34-38] we described the time evolution of the logarithmic negativityEN (t) for two bosonic modes interacting with a common environment. In the present

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case the asymptotic logarithmic negativity is given by

EN (∞) =− log2 coth1

2kTmin, (25)

where Tmin = minT1,T2. It depends on temperatures only, and does not dependon the initial Gaussian state. It takes only negative values, confirming the previousstatement that the asymptotic state is always separable.

4. SUMMARY

In the framework of the theory of open quantum systems based on completelypositive quantum dynamical semigroups, we investigated the Markovian dynamics ofthe quantum entanglement for a subsystem composed of two non-interacting modes,each one embedded in its own thermal bath. By using the Peres-Simon necessary andsufficient condition for separability of two-mode Gaussian states, we have describedthe evolution of entanglement in terms of the covariance matrix for Gaussian inputstates. For all values of the temperatures of the thermal reservoirs, an initial separa-ble non-squeezed Gaussian state remains separable for all times. In the case of anentangled initial Gaussian state (squeezed vacuum state), entanglement suppression(entanglement sudden death) takes place. For definite values of temperatures anddissipation constants, one can observe temporary revivals of the entanglement, butthe system evolves asymptotically to an equilibrium state which is always separable.We calculated the asymptotic logarithmic negativity, which characterizes the degreeof entanglement of the quantum state. It depends only on temperatures and does notdepend on the initial Gaussian state. It takes negative values, confirming the fact thatthe asymptotic state is always separable.

Acknowledgments. The author thanks the organizers for a stimulating and productive workshopand acknowledges the financial support received from the Romanian Ministry of Education and Re-search, through the Projects CNCS-UEFISCDI PN-II-ID-PCE-2011-3-0083 and PN 09 37 01 02/2009.

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