Rényi-2 quantum correlations of two-mode Gaussian systems in a thermal environment

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Phys. Scr. 87 (2013) 038108 (5pp) doi:10.1088/0031-8949/87/03/038108

Renyi-2 quantum correlations oftwo-mode Gaussian systems in athermal environmentAurelian Isar1,2

1 National Institute of Physics and Nuclear Engineering, PO Box MG-6, Bucharest-Magurele, Romania2 Academy of Romanian Scientists, 54 Splaiul Independentei, Bucharest 050094, Romania

E-mail: isar@theory.nipne.ro

Received 10 November 2012Accepted for publication 1 December 2012Published 11 February 2013Online at stacks.iop.org/PhysScr/87/038108

AbstractIn the framework of the theory of open systems based on completely positive quantumdynamical semigroups, we provide a description of continuous variable Gaussian Renyi-2(GR2) quantum discord for a system consisting of two non-interacting non-resonant bosonicmodes embedded in a thermal environment. In the case of both an entangled initial squeezedvacuum state and a squeezed thermal state, GR2 quantum discord has non-negative values forall temperatures of the environment. We describe also the time evolution of GR2 classicalcorrelations and GR2 quantum mutual information of the quantum system. All these quantitiestend asymptotically to zero in the long-time regime under the effect of the thermal bath.

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

(Some figures may appear in colour only in the online journal)

1. Introduction

In recent years, there has been increasing interest inusing non-classical entangled states of continuous variablesystems in applications of quantum information processing,communication and computation [1–3]. In this respect,Gaussian states, in particular two-mode Gaussian states, playa key role since they can be easily created and controlledexperimentally. Due to the unavoidable interaction withthe environment, in order to realistically describe quantuminformation processes it is necessary to take decoherence anddissipation into consideration. Decoherence and dynamics ofquantum entanglement in continuous variable open systemshave been intensively studied over the last few years [4–20].

In quantum information theory, an interesting family ofadditive entropies is represented by Renyi-α entropies [21],defined by

Sα(ρ) = (1 − α)−1 ln Tr(ρα). (1)

In the limit α → 1 they reduce to the von Neumannentropy S(ρ) = −Tr(ρ ln ρ), which in quantum informationtheory quantifies the degree of information contained ina quantum state ρ, and in the case α = 2 we obtain

S2(ρ) = − ln Tr(ρ2), which is a quantity directly related tothe purity of the state. von Neumann entropy satisfies thestrong subadditivity inequality which is a key requirementfor quantum information theory, implying in particular thatthe mutual information, which measures the total correlationsbetween two subsystems in a bipartite state, is alwaysnon-negative. Renyi entropies are powerful quantities forstudying quantum correlations in multipartite states [22, 23].In general, Renyi-α entropies for α 6= 1 are not subadditive.In [24], it was demonstrated that Renyi-2 entropy provides anatural measure of information for any multimode Gaussianstate of quantum harmonic systems. It was proven that for allGaussian states this entropy satisfies the strong subadditivityinequality, which made it possible to define measures ofGaussian Renyi-2 (GR2) entanglement, total, classical anddiscord-like quantum correlations based on this entropy. Inthis sense one could regard Renyi-2 entropy as a speciallymeaningful choice to develop a Gaussian theory of quantuminformation and correlations [24].

In this paper we study, in the framework of the theoryof open systems based on completely positive quantumdynamical semigroups, the dynamics of continuous variableGR2 quantum discord of a subsystem consisting of two

0031-8949/13/038108+05$33.00 Printed in the UK & the USA 1 © 2013 The Royal Swedish Academy of Sciences

Phys. Scr. 87 (2013) 038108 A Isar

bosonic modes (harmonic oscillators) interacting with acommon thermal environment. In section 2, we give thegeneral solution of the evolution equation for the covariancematrix, i.e. we derive the variances and covariances ofcoordinates and momenta. We are interested in discussing thecorrelation effect of the environment; therefore, we assumethat the two modes are independent, i.e. they do not interactdirectly. The initial state of the subsystem is taken to beof Gaussian form and the evolution under the quantumdynamical semigroup assures the preservation in time ofthe Gaussian form of the state. In section 3, we show thatin the case of both an entangled initial squeezed vacuumstate and squeezed thermal state, GR2 quantum discord hasnon-negative values in time for all values of the temperature ofthe environment. We describe also the time evolution of GR2classical correlations and GR2 quantum mutual information,which measures the total amount of GR2 correlations of thequantum system. All these quantities tend asymptotically tozero in time under the effect of the thermal bath. A summaryis given in section 4.

2. Equations of motion for two modes interactingwith an environment

We study the dynamics of a subsystem composed of twonon-interacting bosonic modes in weak interaction witha thermal environment. In the axiomatic formalism basedon completely positive quantum dynamical semigroups, theMarkovian irreversible time evolution of an open systemis described by the Kossakowski–Lindblad master equation[25, 26]. We are interested in the set of Gaussian states;therefore, we introduce such quantum dynamical semigroupsthat preserve this set during the time evolution of the system.The Hamiltonian of the two uncoupled non-resonant harmonicoscillators of identical mass m and frequencies ω1 and ω2 is

H =1

2m(p2

x + p2y) +

m

2(ω2

1x2 + ω22 y2), (2)

where x, y are the coordinates and px , py are the momenta ofthe oscillators.

The equations of motion for the quantum correlations ofthe canonical observables x, y and px , py are the following(T denotes the transposed matrix) [26]:

dσ(t)

dt= Yσ(t) + σ(t)Y T + 2D, (3)

where

Y =

−λ 1/m 0 0

−mω21 −λ 0 0

0 0 −λ 1/m

0 0 −mω22 −λ

,

D =

Dxx Dxpx Dxy Dxpy

Dxpx Dpx px Dypx Dpx py

Dxy Dypx Dyy Dypy

Dxpy Dpx py Dypy Dpy py

, (4)

and the diffusion coefficients Dxx , Dxpx , . . . and the dissi-pation constant λ are real quantities. We introduced thefollowing 4 × 4 bimodal covariance matrix:

σ(t) =

σxx (t) σxpx (t) σxy(t) σxpy (t)

σxpx (t) σpx px (t) σypx (t) σpx py (t)

σxy(t) σypx (t) σyy(t) σypy (t)

σxpy (t) σpx py (t) σypy (t) σpy py (t)

=

(A C

CT B

), (5)

where A, B and C are 2 × 2 Hermitian matrices. Aand B denote the symmetric covariance matrices for theindividual reduced one-mode states, while the matrix Ccontains the cross-correlations between modes. The elementsof the covariance matrix are defined as σi j = Tr[ρ{Ri R j +R j Ri }]/2, i, j = 1, . . . , 4, with R = {x, px , y, py}, which upto local displacements fully characterize any Gaussian stateof a bipartite system [27]. All the measures defined below areinvariant under local unitaries, so we will assume our states tohave zero first moments, 〈R〉 = 0, without loss of generality.

The time-dependent solution of equation (3) is givenby [26]

σ(t) = M(t)[σ(0) − σ(∞)]MT(t) + σ(∞), (6)

where the matrix M(t) = exp(Y t) has to fulfill the conditionlimt→∞M(t) = 0. The values at infinity are obtained from theequation

Yσ(∞) + σ(∞)Y T= −2D. (7)

3. Dynamics of Gaussian Renyi-2 (GR2) correlations

3.1. Time evolution of GR2 quantum discord

The GR2 mutual information I2 for an arbitrary bipartiteGaussian state ρAB is defined by

I2(ρA:B) = S2(ρA) +S2(ρB) −S2(ρAB), (8)

where ρA and ρB are the two marginals of ρAB . It was shownin [24] that I2(ρA:B)> 0 and it measures the total quadraturecorrelations of ρAB .

One can also define a GR2 measure of one-way classicalcorrelations J2(ρA|B) [24, 28] as the maximum decrease inthe Renyi-2 entropy of subsystem A, given that a Gaussianmeasurement has been carried out on subsystem B, where themaximization is over all Gaussian measurements, which mapGaussian states into Gaussian states [29, 30].

Following [31] and the analysis of Gaussian quantumdiscord using von Neumann entropy [29, 30], in [24] Adessoet al defined a Gaussian non-negative measure of quantum-ness of correlations based on Renyi-2 entropy, namely GR2discord, as the difference between mutual information (8) andclassical correlations:

D2(ρA|B) = I2(ρA:B) −J2(ρA|B). (9)

For a general two-mode Gaussian state ρAB , with A andB single modes, closed formulae have been obtained for

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Phys. Scr. 87 (2013) 038108 A Isar

GR2 classical correlations and GR2 discord [24]. For purebipartite Gaussian states ρAB , the following equalities hold:12I2(ρA:B) = J2(ρA|B) = D2(ρA|B) = S2(ρA).

The covariance matrix σ (5) of any two-mode Gaussianstate can be transformed, by means of local unitary operations,into a standard form of the type [32]

σ =

(A C

CT B

)=

1

2

a 0 c+ 0

0 a 0 c−

c+ 0 b 0

0 c− 0 b

, (10)

where a, b > 1, [(a2− 1)(b2

− 1) − ab(c2−

+ c2+) − 2c−c+ +

c2−

c2+]> 0, and we can set c+ > |c−| without losing generality.

These conditions ensure that the uncertainty relation σ >iω⊕2/16 is satisfied (ω =

(0−1

10

)is the symplectic matrix),

which is a requirement for the covariance matrix σ to beassociated with a physical Gaussian state in a two-modeinfinite-dimensional Hilbert space [32]. For pure Gaussianstates, b = a, c+ = −c− =

√a2 − 1.

For generally mixed two-mode Gaussian states ρAB , theRenyi-2 measure of one-way quantum discord (9) has thefollowing expression if the covariance matrix σ is in standardform (10) [24]:

D2(ρA|B) = ln b −12 ln(det C) + 1

2 ln ε2, (11)

where

ε2 =

a(

a −c2

+b

), if (ab2c2

−− c2

+(a + bc2−))

× (ab2c2+ − c2

−(a + bc2

+)) < 0;

2|c−c+|

√(a(b2 − 1) − bc2

−)(a(b2 − 1) − bc2+)

+(a(b2− 1) − bc2

−)(a(b2

− 1) − bc2+) + c2

−c2

+

(b2

− 1)2 ,

otherwise.

(12)

These formulae, written explicitly for standard formcovariance matrices, can be re-obtained in a locally invariantform by expressing them in terms of the four local symplecticinvariants of a generic two-mode Gaussian state [33], I1 =

4 det A, I2 = 4 det B, I3 = 4 det C and I4 = 16 det σ [34].This can be realized by inverting the relations I1 = a2, I2 =

b2, I3 = c+c− and I4 = (ab − c+)(ab − c−) [32]. The obtainedexpressions are then valid for two-mode covariance matricesin any symplectic basis, beyond the standard form.

We assume that the initial Gaussian state is a two-modesqueezed thermal state, with the covariance matrix of theform [35, 36]

σs(0) =1

2

as 0 cs 0

0 as 0 −cs

cs 0 bs 0

0 −cs 0 bs

, (13)

with the matrix elements given by

as = n1 cosh2 r + n2 sinh2 r + 12 cosh 2r, (14)

0

5

10

15

20

t

0

1

2

3

4

T

0.00

0.01

0.02

0.03

D2v

Figure 1. GR2 quantum discord D2v(ρA|B) versus time t andtemperature T for an entangled initial squeezed vacuum statewith squeezing parameter r = 0.23, n1 = 0, n2 = 0 andλ = 0.1, ω1 = 1, ω2 = 2. We take m = h = k = 1.

bs = n1 sinh2 r + n2 cosh2 r + 12 cosh 2r, (15)

cs =12 (n1 + n2 + 1) sinh 2r, (16)

where n1, n2 are the average number of thermal photonsassociated with the two modes and r denotes the squeezingparameter. In the particular case n1 = 0 and n2 = 0, (13)becomes the covariance matrix of the two-mode squeezedvacuum state [37]. A two-mode squeezed thermal stateis entangled when the squeezing parameter r satisfies theinequality r > rs [35], where

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

n1 + n2 + 1. (17)

We suppose that the asymptotic state of the consideredopen system is a Gibbs state corresponding to twoindependent bosonic modes in thermal equilibrium attemperature T [38]. Then the quantum diffusion coefficientshave the following form (we put h = 1) [25]:

mω1 Dxx =Dpx px

mω1=

λ

2coth

ω1

2kT,

mω2 Dyy =Dpy py

mω2=

λ

2coth

ω2

2kT, (18)

Dxpx = Dypy = Dxy = Dpx py = Dxpy = Dypx = 0.

The evolution of the GR2 quantum discord is illustratedin figures 1 and 2, where we represent the dependenceof D2(ρA|B) on time t and temperature T for an initialentangled Gaussian state with the covariance matrix given byequation (13), taken as of the form of a two-mode squeezedvacuum state, respectively a squeezed thermal state, and forsuch values of the parameters that satisfy for all times thefirst condition in formula (12). The GR2 discord has positivevalues for all finite times, and in the limit of infinite timeit tends asymptotically to zero, corresponding to the thermal

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Phys. Scr. 87 (2013) 038108 A Isar

0

5

10

15

20

t

0

1

2

3

4

T

0.000

0.002

0.004

0.006

0.008D2

Figure 2. GR2 quantum discord D2(ρA|B) versus time t andtemperature T for an entangled initial non-symmetric squeezedthermal state with squeezing parameter r = 0.1, n1 = 1, n2 = 0and λ = 0.1, ω1 = 1, ω2 = 2. We take m = h = k = 1.

0

5

10

15

20

t

0

1

2

3

4

T

0.00

0.02

0.04

J2

Figure 3. GR2 classical correlations J2(ρA|B) versus time t andtemperature T for an entangled initial non-symmetric squeezedthermal state with squeezing parameter r = 0.1, n1 = 1, n2 = 0and λ = 0.1, ω1 = 1, ω2 = 2. We take m = h = k = 1.

product (separable) state, with no correlations at all. We alsonote that the decay of GR2 discord becomes stronger as thetemperature T and dissipation constant λ increase.

3.2. GR2 classical correlations and quantum mutualinformation

For a mixed two-mode Gaussian state ρAB , GR2 measureof one-way classical correlations J2(ρA|B) has the followingexpression if the covariance matrix σ is in standard form(10) [24]:

J2(ρA|B) = ln a −12 ln ε2, (19)

where ε2 is given by equation (12), while the expression forthe GR2 quantum mutual information, which measures thetotal correlations, is given by

I2(ρA:B) = ln a + ln b −12 ln(det C). (20)

0

5

10

15

20

t

0

1

2

3

4

T

0.00

0.02

0.04

I2

Figure 4. GR2 quantum mutual information I2(ρA:B) versus time tand temperature T for an entangled initial non-symmetric squeezedthermal state with squeezing parameter r = 0.1, n1 = 1, n2 = 0 andλ = 0.1, ω1 = 1, ω2 = 2. We take m = h = k = 1.

0

5

10

t

0.0

0.5

1.0

1.5

2.0

T

0.00

0.02

0.04

0.06

I2,J2,D2

Figure 5. GR2 quantum mutual information, classical correlationsand quantum discord versus time t and temperature T for anentangled initial non-symmetric squeezed thermal state withsqueezing parameter r = 0.1, n1 = 1, n2 = 0 and λ = 0.1,ω1 = 1, ω2 = 2. We take m = h = k = 1.

In figures 3 and 4, we illustrate the evolution of classicalcorrelations J2(ρA|B) and, respectively, quantum mutualinformation I2(ρA:B) as functions of time t and temperatureT for an entangled initial Gaussian state, taken as of theform of a two-mode squeezed thermal state (13), and forsuch values of the parameters that satisfy for all times thefirst condition in formula (12). Both these quantities manifesta similar qualitative behavior: they have positive valuesfor all finite times, and in the limit of infinite time theytend asymptotically to zero, corresponding to the thermalproduct (separable) state, with no correlations at all. One canalso see that the classical correlations and quantum mutualinformation decrease with increasing the temperature of thethermal bath and the dissipation coefficient. For comparison,GR2 mutual information, classical correlations and discordare represented also on the same graphic in figure 5. In theconsidered case, the value of GR2 classical correlations is

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Phys. Scr. 87 (2013) 038108 A Isar

larger than that of quantum correlations, represented by theGR2 quantum discord.

4. Summary

We investigated the Markovian dynamics of GR2 quantumcorrelations for a subsystem composed of two non-interactingbosonic modes embedded in a thermal bath. We have analyzedthe influence of the environment on the dynamics of GR2quantum discord, classical correlations and quantum mutualinformation, which measures the total GR2 correlationsof the quantum system, for initial squeezed vacuumstates and non-symmetric squeezed thermal states, for thecase when the asymptotic state of the considered opensystem is a Gibbs state corresponding to two independentquantum harmonic oscillators in thermal equilibrium.The dynamics of these quantities depend strongly onthe initial states and the parameters characterizing theenvironment (the dissipation coefficient and temperature).Their values decrease asymptotically in time with increasingthe temperature and dissipation.

The study of time evolution of these measures maycontribute to the understanding of the quantification ofcorrelations defined on the basis of the Renyi-2 entropy,which could represent a special tool for developing a Gaussiantheory of quantum information and correlations.

Acknowledgments

The author acknowledges financial support from theRomanian Ministry of Education and Research through theprojects CNCS-UEFISCDI PN-II-ID-PCE-2011-3-0083 andPN 09 37 01 02/2010.

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