Dynamic of the Gaussian quantum discord and effect of

non-Markovian degree

Xin LIU , Wei WU and Changkui HU

Department of Physics , School of Science, Wuhan University of Technology（WHUT）,Wuhan, China

We study the dynamic of the Gaussian quantum discord in a continuous-variable system subject to a

common non-Markovian environment with zero-temperature. By considering an initial two-mode Gaussian

symmetric squeezed thermal state（STS）, we show Gaussian discord has a very different dynamic characteristic in

a non-Markovian evolution versus a Markov process. It can be created by the memory effect which features

non-Markovianity. We also study the relationship between Gaussian discord and non-Markovian degree of

environment. The results may offer us an effective method in experiments to get more quantum correlations.

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

Ⅰ. INTRODUCTION

Quantum correlations(QCs) which have very different features versus classical correlations are

a fundamental resource for quantum information processing(QIP)[1]. In the early years people

belief that entanglement can characterize QCs and gives quantum computers an advantage over

their classical counterparts. However with some new discovery people found that entanglement is

just only one kind of QCs and can not represent all nonclassical correlations[2]. Now there are

several measures have been proposed to detect QCs, a widely acceptable concept is called

quantum discord(QD)[3] which was introduced by Ollivier and Zurek. The QD was defined as the

mismatch between two natural quantum extensions of classically equivalent expression of the

mutual information and it capture a fundamental feature in a bipartite state.

We have known that QD has a significant application in deterministic quantum computation

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with one pure qubit(DQC1)[4,5] and can be used to better understand the quantum phase

transition[6,7]. It can even be used to define the class of initial system-bath states for which the

quantum dynamic is equivalent to a completely positive map[8], and to measure the QCs between

relatively accelerated observers[9]. The concept of discord has been extended to the domain of

bipartite quantum continuous-variable(CV) systems where it is called Gaussian quantum

discord(GQD) in recently years[10-11].

In literatures dynamic of open quantum system[12,13] is often considered in a Markov

process which is described by a master equation for the reduced density matrix with Lindblad

structure[14,15] in theory. However, Lindblad master equation is derived just under a number of

drastic simplifications in realistic physical system. So recently many methods have been devoted

to define non-Markovianity of an open quantum system[16,17] and to quantify the degree of

non-Markovian behaviors[18-20].

In a discrete system some important results for QD in non-Markovian evolution have been

mentioned in Ref[21,22]. It is natural to wonder how these results are generalized for a quantum

CV system which routinely occurs in quantum optics. But now few discussions have been made

for GQD in a non-Markovian environment[23]. So our goal in this paper is to analyze dynamic of

GQD in a CV system of a non-Markovian process. By evaluating the evolution of GQD we find

that GQD can be created by memory effect which is different from a Markovian case. The

relationship between GQD and the degree of non-Marmovianity is also investigated.

The paper is structured as follows. In Sec. Ⅱ we review the definition of GQD and its

expression in STS. In Sec. Ⅲ we describe a definition of non-Markovianity for CV system which

we use in this paper. In Sec. Ⅳ we study dynamic of GQD of a two-mode Gaussian symmetric

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STS and the effect of non-Markovian degree. Finally, In Sec. Ⅴ we draw our conclusions.

Ⅱ. DEFINITION OF GQD IN TWO-MODE GAUSSIAN STATES

The quantum mutual information between two systems A and B in a state ABρ is defined as

)()()()( ABBAAB SSSI ρρρρ −+= (1)

where )log(Tr)( 2 ρρρ −=S is the von Neumann entropy of an arbitrary state ρ and

ABB(ABA ρρ ))( Tr= is the reduced density operator which is describing the state of A(B). An

other expression for the mutual information which is defined via a measurement-based conditional

density operator is

})(Sp)(S{max)(Ji iAiA}{ABA i ∑−= ∏ ρρρ (2)

where }{ i∏ denotes a complete set of positive operator-valued measure(POVM) which is

performed on the subsystem B and iiABBiAp/)(Tr ∏= ρρ is the remaining state of

subsystem A after the measurement with probability )(Tr iABABip ∏= ρ . The mismatch

between Eq.(1) and Eq.(2) is defined as A QD[2,3]

)()()( ABAABABA JID ρρρ −= (3)

Analogously, B QD is defined where the roles of A and B are swapped.

Now a Gaussian version of QD is defined[10,11] in an bipartite CV system. We restrict ABρ

to a two-mode Gaussian state whose covariance matrix can be transformed in a standard form

)(BC

CATAB =σ (4)

with ),,(diag),,(diag bbBaaA == and ),(diag dcC = . The quantities AI det1 = ,

BI det2 = , CI det3 = , and σdet4 =I are referred to as symplectic invariants for they are

invariant under local symplectic transformations. Because of )( ABAJ ρ and )( ABAD ρ are

invariant under local unitaries[2,3], we can extend A QD to a general two-mode Gaussian

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state[10,11] based on the minimization of the mismatch )()( ABAAB JI ρρ − which are over

single-mode generalized Gaussian measurements on subsystem B as follow

)det(inf)()()()( 2 AABA hdhdhIhCM

σρσ

+−−= +− (5)

where )2

1ln()

2

1()

2

1ln()

2

1()( −−−++= xxxxxh , ±d are symplectic eigenvalues of ABρ

and can be expressed by 2

4 4

2

2I

d−∆±∆

=±

with 321 2III ++=∆ , Mσ is the

covariance matrix of the single-mode Gaussian states which are performed for Gaussian

measurements on subsystem B, T

MA CBCA 1)( −+−= σσ [25] is the covariance matrix of

the subsystem A after a measurement. In this paper, we will focus on the relevant subclass of

two-mode STSs with a generic state

)()( 21 rVrV +⊗= ρρρ (6)

where )()( abbarerV −++

= is the two-mode squeezing operator and )2,1( =iiρ is given by

||)1( 1 nnNN i

n

in

n

ii <>+= −−∑ρ with iN is the average number of thermal photons. For

a case of STS, we can obtain the diagonal matrix elements defining A, B, and C

rrr NNNNNa 21 )1()2

1( ++++= (7)

rrr NNNNNb 12 )1()2

1( ++++= (8)

)1()1( 21 rr NNNNdc +++=−= (9)

where rN r

2sinh= . Gaussian A QD for a generic bipartite STS then can be written as

)21

22()()()()(

2

3211

2I

IIIIhdhdhIhC ABA

+

+++−−= +−ρ (10)

By exchanging 21 II ↔ , Gaussian B QD is obtained with Gaussian measurements on subsystem

A. In the following parts, we will use Gaussian A QD as GQD.

Ⅲ. NON-MARKOVIAN DEGREE IN A CONTINUOUS-VARIABLE SYSTEM

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In order to detect non-Markovian degree of a quantum system, many measures have been

presented [16-20]. In this paper we use the measurement which was first introduced by H. P.

Breuer et al[8] and it was extended to a CV system by R. Vasile et al[24]. In this measurement the

information flows between an open system and its environment is treated as the key feature of

non-Markovianity.

The fundamental idea of the measure is that a Markovian process tends to continuously

reduce the distinguishability between any two states, contrarily the increasing of the

distinguishability means non-Markovianity. The interpretation of this phenomenon is that the loss

of distinguishabilty of two states is because of a flow of information which is from the open

system to its environment and when it reverses, the distinguishability arises. The measure is based

on the trace distance[1] of two quantum states which describes the probability of successfully

distinguishing the states. The trace distance is defined as

2121 tr2

1),( ρρρρ −=D (11)

where AAA += . In a Markovian dynamic the trace distance is always monotonic decrease,

namely

))(),(())(),(( 2121 ttDttD ρρτρτρ ≤++ (12)

When the monotonicity is not satisfied, it means that there are intervals of time for which the two

states become more distinguishable compared to previous instants and it then characterizes a

non-Markovian evolution. In a generic CV state, there has no analytic expression for the trace

distance but an alternative signature can be employed within the same spirit such as the fidelity

which is defined as follow

12121 tr),( ρρρρρ =F (13)

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It is a good candidate for taking the role of the trace distance in the definition of the

non-Markovian measure. That is in a CV system,

))(),(())(),(( 2121 ttFttF ρρτρτρ ≤++ (14)

holds for a Markovian process; when it fails, there defines a non-Markovian process.

In this paper we consider the dynamic which is described by the phenomenological equation

with a single decay channel in Lindblad form

)2(2

)(aaaaaa

t

dt

d +++ −−= ρρργ

αρ

(15)

Where a is the annihilation operator and )(tγ is a time-dependent damping rate. The function

)(tγ is determined by the spectral density )(wJ of the reservoir. Any Gaussian state which is

evolving according to Eq. (15) remains Gaussian, and the covariance matrix in zero-temperature

environment(the average number of thermal photons for environment 0=eN ) evolves as

follow[26]

2]1[)0()( )()( Ι

−+= −− txtx eet σσ (16)

where ∫=t

dsstx0

)(2)( γα and α is a coupling constant. It has been proved[18,24] that Eq.

(12) or (14) is not satisfied whenever 0)( <tγ . Then we can say the evolution is

non-Markovianity. So Eq. (15) describes a non-Markovian process if there exists some negative

intervals of )(tγ and it means Eq. (14) fails for certain times in a physics process. The degree of

non –Markovianity can be obtained by integrating the time derivatives of the fidelity over the

intervals in which it decreases. It can be written as follow

]),P([max0P

dttFdt

dN

Fd ∫ <

⋅−= (17)

where symbol P denotes a full set of parameters and⋅

F indicates the time derivatives, the

maximization is taken over the set of parameters P.

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Ⅳ. BEHAVIOR OF GAUSSIAN DISCORD AND EFFECT OF NON-MARKOVIAN DEGREE

In this section we study the behavior of GQD of a two-mode Gaussian STS in a common

non-Markovian reservoir with zero-temperature which is described in the previous section. We

analyze a case of symmetric STS in which 21 NN = . In example, we consider the damping rate

is

≥

<×=

−

−

2/5,

2/5,sin

2

1)(

2/

5/

ππ

γπ tife

tiftet

t

(18)

which is characterized by only one interval of negativity, ]2,[ ππ for non-Markovianity,

1021 == NN and 2=r . We plot the evolution of GQD with 5.0=α in Fig. 1(b). In

contrast, the situation for a Markovian process is exhibited in Fig. 1(a) with the damping rate is

≥

<×=

−

−

ππ

γπ tife

tiftet

t

,

,sin

2

1)(

2/

5/

(19)

(a) (b)

Fig. 1. Evolution of GQD in (a) a Markov case and in (b) a non-Markovian case with

1021 == NN 、 2=r and 5.0=α . Inset: zoom on small values of GQD.

We can see that GQD monotonically decreases in a Markov process, but it is very different in

a non-Markovian case. During the decay GQD has a rise suddenly and then decreases to zero

eventually. By observing the rising time in details, we find it is just the time when )(tγ is

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negative. Then we adjust the damping rate as follow form

≥

<×=

−

−

2/9,

2/9,sin

2

1)(

2/

5/

ππ

γπ tife

tiftet

t

(20)

which has two intervals of negativity, ]2,[ ππ and ]4,3[ ππ . In Fig. 2 we can see there are two

rises for GQD during the decrease and both the happening time are the time when )(tγ is

negative. This feature is not specific to the two cases which we have chosen but occurs for all

non-Markovian cases when there exists time intervals for negative )(tγ . As we have mentioned

before, a Markovian process tend to continuously reduce the distinguishability between any two

states, while the essential property of non-Markovian behavior is a growth of this

distinguishability. Once 0)( <tγ , Eq.(14) fails and it means the distinguishability is growing

and we can interpret this growth as a backflow of information which is from environment. Just

because of the backflow information, recoherence occurs and quantum correlations are created in

a non-Markovian process. From the definition in former section, we notice that GQD is positive as

far as 0≠rN . So bipartite Gaussian states have always nonzero GQD except when they are

product states. We also find since GQD is a decreasing function of a and b and an increasing

function of c , it will decrease to zero eventually in noise channels with the time evolution. If one

wants to know the exact quantity of GQD at an exact time, he needs to find the relevant

parameters and calculate it. By using the result we have obtained above, there is a way to find

whether GQD is increasing in practice. That is in an experiment one can perform a state

tomography for two states of the open system at different moments so as to decide whether the

fidelity has increased. If it happens, it means GQD has been created. Since it has been proved that

the effect of non-Markovianity can produce oscillations in the dynamic of Gaussian

entanglement[17], we will study the relationship between the evolution of discord and that of

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entanglement in further work.

Fig. 2. Evolution of GQD with 1021 == NN 、 2=r 、 5.0=α

and )(tγ is in the form of Eq. (20). Inset: zoom on small values of the GQD.

For quantum correlations can be created due to non-Markovianity of the system, it is natural

to wonder how is the influence of the degree of non-Markovianity. In a symmetric STS it has been

shown that the degree of non-Markovianity, as defined by Eq. (17), can be simplified as[24]

dttrk

rkrkrNd )(2

),(

),(),()2cosh(8

02γ

φφφ

αγ∫ <⋅⋅

−⋅= (21)

where )cos1(4coshcos3),( φφφ −++= rrk and φ is the angle between the squeezing

directions. The expression is valid for any form of the damping rate )(tγ and it is

straightforward to show that this result is independent of the number of negativity periods of

)(tγ . For the sake of concreteness, let us consider the example which is mentioned above with the

damping rate is in form of Eq.(18). Now we start the analysis on evaluating the non-Markovianity.

In Fig. 3(a) it can be seen that the degree of non-Markovianity is proportional to the coupling

constant. We show the relationship between GQD and the degree of non-Markovianity at different

time in Fig. 4(b). At an exact time, GQD decreases with increasing dN . But at different time

when )(tγ is negative, GQD increases at a same dN which corresponds to the result we have

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got above.

(a) (b)

Fig. 3. (a)Non-Markovian degree as a function of the coupling α . (b) GQD as a function of

non-Markovian degree dN at different time.

Then we focus on the feature of relationship between GQD and non-Markovian degree under

time evolution. In Fig. 4(a) we can see that when the degree of non-Markovianity is lower, the rise

of GQD is less and GQD decreases more smooth and slow. Physically, it can be interpreted as

more information will flow from system to environment with higher non-Markovian degree. So

when it reverses during the period of negative )(tγ , more information flows back and more GQD

is created. Namely memory effect is stronger with higher degree of non-Markovianity. It may offer

us an effective method in experiments to get more quantum correlations according to control the

non-Markovian degree of environment. The evolution as a function of non-Markovian degree in

three-dimension is shown in Fig. 4(b). It shows the results we have got more clearly.

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Fig. 4. The evolution of GQD (a) with different non-Markovianity degree and (b) as a function

of non-Markovianity in three-dimension.

Ⅴ. CONCLUSIONS

In this paper, we have studied the non-Markovian effect on the dynamic of GQD in a CV

system. By showing the behavior of GQD of a two-mode Gaussian symmetric STS in a common

non-Markovian reservoir, we found that in a non-Markovian process GQD rises during the time

whenever damping rate is negative. It means quantum correlations can be created in

non-Markovian reservoir. We have given a physical explanation to this phenomenon. We also have

found the relationship between the increasing GQD and the degree of non-Markovianity. It will be

very significant for practical purpose.

ACKNOWLEDGMENTS

This work was supported by “the Fundamental Research Funds for the Central Universities”

under the project number 2013-Ia-032 and “the Fundamental Research Funds for the Central

Universities(WUT:2014-Ia-026)” .

REFERENCES

[1] M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge

University Press, Cambridge, UK, 2000).

Page 11 of 13C

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Y B

RO

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7/14

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use

onl

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his

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-IN

man

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or to

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iting

and

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rd.

[2] L. Henderson and V. Vedral, J. Phys. A 34, 6899(2001);V. Vedral, Phys. Rev. Lett. 90,

050401(2003).

[3] H. Ollivier and W. H. Zurk, Phys. Rev. Lett. 88, 017901(2001).

[4] T. Yu and J.H. Eberly, Science 323, 598(2009).

[5] A. Datta, A. Shaji, and C.M. Caves, Phys. Rev. Lett.100, 050502(2008).

[6] R. Dillenschneider, Phys. Rev. B 78, 224413(2008).

[7] M. S. Sarandy, Phys. Rev. A 80, 022108(2009).

[8] A. Shabani and D. A. Lidar, Phys. Rev. Lett. 102, 100402(2009).

[9] A. Datta, Phys. Rev. A 80, 052304(2009).

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

[11] G. Adesso and A. Datta, Phys. Rev. Lett. 105, 030501(2010).

[12] H. P. Breuer and F. Petruccione, The Theory of Open Quantum Systems (Oxford University

Press, Oxford, 2007).

[13] U. Weiss, Quantum Dissipative Systems, 3rd ed. (World Scientific, Singapore, 2008).

[14] V. Gorini, A. Kossakowski, and E. C. G. Sudarshan, J. Math. Phys. (N.Y.)17,821(1976).

[15] G.Lindblad, Commun. Math. Phys. 48, 119(1976).

[16] L. Mazzola, S. Maniscalco, J. Piilo, K. A. Suominen, and B. M. Garraway, Phys. Rev. A 80,

012104(2009).

[17] S. Maniscalco, S. Olivares, and M. G. A. Paris, Phys. Rev. A 75, 062119(2007); R. Vasile, S.

Olivares, M. G. A. Paris, and S. Maniscalco, ibid. 80, 062324(2009).

[18] H. P. Breuer, E. M. Laine, and J. Piilo, Phys. Rev. Lett. 103, 210401(2009); E. M. Laine, J.

Piilo, and H. P. Breuer, Phys. Rev. A 81, 062115(2010).

Page 12 of 13C

an. J

. Phy

s. D

ownl

oade

d fr

om w

ww

.nrc

rese

arch

pres

s.co

m b

y SU

NY

AT

ST

ON

Y B

RO

OK

on

10/0

7/14

For

pers

onal

use

onl

y. T

his

Just

-IN

man

uscr

ipt i

s th

e ac

cept

ed m

anus

crip

t pri

or to

cop

y ed

iting

and

pag

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n. I

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fer

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the

fina

l off

icia

l ver

sion

of

reco

rd.

[19] M. M. Wolf, J. Eisert, T. S. Cubitt, and J. I. Cirac, Phys. Rev. Lett. 101, 150402(2008).

[20] A. Rivas, S. F. Huelga, and M. B. Plenio, Phys. Rev. Lett. 105, 050403(2010).

[21] B. Wang, Z.-Y. Xu, Z.-Q. Chen, and M. Feng, Phys. Rev. A 81, 014101(2010).

[22] F. F. Fanchini, T. Werlang, C. A. Brasil, L. G. E. Arruda, and A. O. Caldeira, Phys. Rev. A 81,

052107(2010).

[23] X. Liu and W. Wu, Chin. Phys. B 23(7), 070303(2014).

[24] R. Vasile, S. Maniscalco, M. G. A. Paris, H. P. Breuer, and J. Piilo, Phys. Rev. A 84,

052118(2011).

[25] J. Eisert and M. Plenio, Int. J. Quantum. Inform. 1, 479(2003); M. Takeoka and M. Sasaki,

Phys. Rev. A 78, 022320(2008).

[26] A. Ferraro, S. Olivares, and M. G. A. Paris, Gaussian States in Quantum

Information(Bibliopolis, Napoli, 2005).

Page 13 of 13C

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