ZHANG Zhan-jun, RUI Pin-shu, WANG Sheng-fang, YE Biao-liang, LIU Dao-chu

(School of Physics & Material Science,Anhui University, Hefei 230039, China)

0 Introduction

Correlation is ubiquitous in the universe. Much attention had been paid on it very early. Since quantum mechanics was established in the last century, the discussion of correlation was extended to the quantum aspect besides its classical feature. People started to realize that correlation includes both classical and quantum parts. However, for a quite long time people mistakenly believed that quantum entanglement, which has exhibited the non-classicality of correlation, completely characterizes and is equivalent to quantum correlation. It is found that entanglement plays crucial roles in many quantum information proceedings[1-12]. Because of this, much attention has been focused on quantum entanglement. However, it has recently been recognized that entanglement can not account for all the nonclassical properties of quantum correlations[13]. Alternatively, quantum entanglement can not represent for quantum correlation fully. Specifically, in some separable systems, where quantum entanglements do not exist at all, nonclassical correlations occur indeed. Due to this newly finding, quantum correlation beyond entanglement has attracted much attention nowadays. It has been found that some quantum tasks can be accomplished via quantum correlation beyond entanglement, such as quantum state merging[14], assisted optimal state discrimination[15], quantum computing[16], etc. The successful revealment of the essential role in those proceedings further stimulates the study of this new kind of quantum correlation recently[13,17-39].

By now, many new methods have been put forward to characterize and quantify quantum correlation[13,17-24]. The first approach is the well-known one named quantum discord proposed by Ollivier and Zurek et al[13]in 2002, where non-classicality is defined as the difference between the total correlation and the measured maximal classical one. In the approach, an optimal measurement should be performed. Otherwise, the measured classical correlation is non-maximal. In principle, the approach of quantum discord is applicable for any bipartite state to extract its classical correlation. Nevertheless, the optimization procedure is actually quite difficult as far as a general state is concerned, and hence only a few of states have been studied so far. Later, in 2008 Luo[18]proposed a new method named measurement induced disturbance (MID), with which non-classicality can be finally extracted via peculiar measurements. It is an easily computable method, for the peculiar measuring bases are exactly the eigenstates of marginal states. The non-classicality captured in such way is referred to as measure-induced non-classicality (MINC) in literatures. Due to its convenience in use, quite many works employed the MID method to estimate MINC in different quantum states[25-30]. Moreover, some other new methods have been put forward to characterize and quantify quantum correlations in various states (not limited to bipartite states anymore), too[31-39]. Here we do not mention them anymore.

In this paper we will employ the famous MID method[18]to study a family of bipartite mixed states we concerned. The states read

ρAB=c0|00〉A(chǔ)B〈00|+c0|11〉A(chǔ)B〈11|+(1-c0-c1)|u2〉A(chǔ)B〈u2|,

(1)

1 Measure-induced non-classicality in the concerned states

Before presenting our study, let us briefly introduce the MID method proposed by Luo in 2008[18]. The basic idea in MID is that, the classical correlation in a bipartite state is captured in the way that the eigenstates of marginal states are taken as measuring bases to measure corresponding subsystems. Such peculiar measurements are based on the so-called spectrum resolution technique in usual. Consider a stateρof a quantum system consisting of subsystemsAandB. The quantum mutual information of the bipartite system in the stateρABis defined as

I(ρAB)=S(ρA)+S(ρB)-S(ρAB),

(2)

whereS(·) represents von Neumann entropy,ρAandρBare marginal states ofρAB. Within the framework of the MID approach, this quantity is taken as the total correlation in the stateρAB. By measuring the subsystemAandB, one can get classical correlation inρAB. As mentioned before, the spectrum resolution technique is adopted by the MID approach. For the two reduced statesρAandρB, their spectrum resolutions are actually treated as

(3)

(4)

C(ρAB)=I(ηρAB)=S(ρA)+S(ρB)-S(ηρAB).

(5)

After the spectrum resolutions, it is very easy to work out the classical correlation in the stateρAB. Alternatively, the classical correlation in the state has been captured via measurements. Meanwhile, from another angle of view one can say that, the non-classicality induced also by measurements has been exposed. By virtue of the MID approach, the MINC of the bipartite stateρABis defined as the difference between the quantum mutual information ofρAB(the total correlation) and that ofηρAB(the classical correlation), i.e.

Q(ρAB)≡I(ρAB)-C(ρAB)=S(ηρAB)-S(ρAB).

(6)

Now let us move to present our study in terms of the MID approach described just. Using Eq.(2) one can get the total correlation in any of our concerned states described by Eq.(1). To be specific

(7)

Fig.1 displays the total correlation ofρABas a function of coefficientsc0andc1, which characterize the concerned states.

Fig.1 Total correlations in our concerned states

Using the spectrum resolution technique, one can rewrite the two reduced statesρAandρBas

(8)

ρB=trAρAB=c0|0〉〈0|+c1|1〉〈1|+(1-c0-c1)|2〉〈2|,

(9)

(10)

After the spectrum resolutions, the measuring bases on either subsystem are actually determined. Then the corresponding measurements on individual subsystems induce the collapse of the considered state. Specifically, the considered stateρABevolves to its classical state

(11)

Note that in the above classical state, the occurrence probability of each component is essentially a function ofc0andc1.

In terms of the definition of classical correlation given by Eq.(5), one is readily to get

(1+w)log2(1+w)]+2(1+cos2θ)(c0+c1)log2cosθ+

2(1-cos 2θ)(c0+c1)log2sinθ+c0+c1.

(12)

Obviously, it is actually a function ofc0andc1, too. Classical correlations ofρABversus the two coefficientsc0andc1are plotted in Fig.2.

Fig.2 Classical correlations captured via the MID method

For a given stateρAB, obviously its total correlation is certain. Since its classical correlation can be captured via the MID method, then its inherent quantum correlation can be consistently retrieved with respect to the definition given by Eq.(6), i.e.

(1-sin 2θ)log2(1-sin 2θ)]-(1+cos 2θ)(c0+c1)log2θ-

(1-cos 2θ)(c0+c1)log2θ+c0+c1-1.

(13)

This is exactly the so-called MINC in the stateρAB. Fig.3 shows its variance with bothc0andc1.Fig. 4 is the contour of Fig.3.

Fig.3 MINC in the concerned states

Fig.4 The contour of Fig.3

2 Discussions

Now let us make some discussions on various correlations in the concerned states and simply analyze them.

(1) From Fig. 1 it is easy to see that the total correlation first increases and then moves to decrease with increasingc1whenc0is given. Such variance also occurs whenc0permutes withc1. Within the family, the state withc0=c1=1/2 has the maximal total correlation, that is, its total correlation equals to 1. From Eq.(1) one is readily to find that the state is actually a classical separable state. The states withc0=c1=0,c0=0 andc1=1 orc0=1 andc1=0 have the minimal total correlation, which is equal to zero. Also from Eq.(1) one can find they are classical product states. Moreover, it is easy to see that the total correlations are symmetric about the linec0=c1.

(2) The detailed variance of the captured classical correlations as a function ofc0andc1is a little complicated. However, from Fig.2 one can find that, the captured classical correlation reaches its maximal value (i.e. 1) whenc0=c1=1/2 and its minimal value (i.e., 0) whenc0=c1=0,c0=0 andc1=1, orc0=1 andc1=0. In fact, these extreme values can be easily understood. In item (1), it has been revealed that the states withc0=c1=1/2,c0=c1=0,c0=0 andc1=1, orc0=1 andc1=0 are all classical states. In these cases, their classical correlations are surely equal to their total correlations. Besides, the same as the symmetry in Fig.1, the classical correlations are symmetric about the linec0=c1, too.

(3) Whenc0(orc1) is set,Qfirst increases and then moves to decrease with increasingc1(orc0). As can be seen from Fig.3, the value ranges from 0 to 0.900. The state with the maximal MINC is

ρAB=0.45|00〉A(chǔ)B〈00|+0.45|11〉A(chǔ)B〈11|+0.1|u2〉A(chǔ)B〈u2|.

Importantly, one can see that, the same as some separable qubit states, some of our concerned separable qubit-qutrit correlated states own quantum correlations, too. Moreover, as mentioned before, both the total and the classical correlations are symmetric about the linec0=c1. Hence, the MINCs as the difference between them are naturally symmetric about the linec0=c1, too.

(4) Figs.(1-3) have exhibited a common feature that there exists the axial symmetry about the beelinec0=c1. Hence there must be an essential reason for the phenomena. Easily, one can verify that

(14)

3 Summary

To summarize, in this paper we have studied the correlations of a family of bipartite separable qutrit-qubit correlated states with the MID method. By tedious deductions we have got the analytic expressions of the total, classical and quantum correlations of the concerned states. For intuition, we have plotted them as functions of the two parameters characterizing the states in the family we concerned. Besides, we have found that in some qubit-qutrit states there also exist quantum correlations. Moreover, we have made some brief discussions on various correlations including MINCs and some of their distinct features are revealed.

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