Quantum discord for two-qubit X-states
Quantum discord, a kind of quantum correlation, is defined as the difference between quantum mutual information and classical correlation in a bipartite system. In general, this correlation is different from entanglement, and quantum discord may be nonzero even for certain separable states. Even in the simple case of bipartite quantum systems, this different kind of quantum correlation has interesting and significant applications in quantum information processing. So far, quantum discord has been calculated explicitly only for a rather limited set of two-qubit quantum states and expressions for more general quantum states are not known. In this paper, we derive explicit expressions for quantum discord for a larger class of two-qubit states, namely, a seven-parameter family of so called X-states that have been of interest in a variety of contexts in the field. We also study the relation between quantum discord, classical correlation, and entanglement for a number of two-qubit states to demonstrate that they are independent measures of correlation with no simple relative ordering between them.
For a given bipartite quantum state, it is important to know whether it is entangled, separable, classically correlated or quantum correlated. Much effort has been invested in subdividing quantum states into separable and entangled states (see Refs. Alber-QI2001 (); Horodecki-RMP-2009 () and references therein). It is well known that entanglement makes possible tasks in quantum information which are impossible without it NC-QIQC-2000 (). However, entanglement is not the only type of correlation useful for quantum technology. Recently, it was found that there are some quantum correlations other than entanglement that also offer some advantage, for example, quantum non-locality without entanglement Bennett-PRA59-1999 (); Horodecki-PRA71-2005 (); Niset-PRA74-2006 (). In addition, it was shown theoretically Braunstein-PRL83-1999 (); Meyer-PRL85-2000 (); Datta-PRL100-2008 (), and later experimentally Lanyon-PRL101-2008 (), that some separable states may also speed up certain tasks over their classical counterparts. Therefore, it is desirable to investigate, characterize, and quantify quantum correlations more broadly.
A bipartite quantum state contains both classical and quantum correlation. These correlations are quantified jointly by their ‘quantum mutual information’, an information-theoretic measure of the total correlation in a bipartite quantum state Groisman-PRA72-2005 (). In particular, if denotes the density operator of a composite bipartite system , and () the density operator of part (), respectively, then the quantum mutual information is defined as
where is the von Neumann entropy. Moreover, it was shown that quantum mutual information is the maximum amount of information that A(lice) can send securely to B(ob) if a composite correlated quantum state is used as the key for a one-time pad cryptographic system Schumacher-PRA74-2006 ().
Quantum mutual information may be written as a sum of classical correlation and quantum correlation , that is, Ollivier-PRL88-2001 (); Vedral-et-al (); Luo-PRA77-2008 (). This quantum part has been called quantum discord Ollivier-PRL88-2001 (). It is a different type of quantum correlation than entanglement because separable mixed states (that is, with no entanglement) can have non-zero quantum discord.
Quantum discord is not always larger than entanglement Luo-PRA77-2008 (); Li-Luo-PRA-2008 (). This indicates that discord is not simply a sum of entanglement and some other nonclassical correlation. Even for the simplest case of two entangled qubits, the relation between quantum discord, entanglement, and classical correlation is not yet clear. For pure states and, surprisingly, for a mixture of Bell states, quantum correlation is exactly equal to entanglement whereas classical correlation attains its maximum value . However, for general two-qubit mixed states, the situation is more complicated. Qubit-qubit entanglement has been characterized and quantified completely whereas quantum discord only for particular cases Ollivier-PRL88-2001 (); Vedral-et-al (); Luo-PRA77-2008 (); Li-Luo-PRA-2008 (); Oppenheim-PRL89-2002 (); Kaszlikowski-PRL101-2008 (); Dillenschneider-PRB78-2008 (); Sarandy-arXiv (). Quantum discord is a measure of nonclassical correlations that may include entanglement but is an independent measure. We will document with simple examples that the amounts of classical correlation, quantum discord and entanglement bear no simple relationship to each other.
In order to quantify quantum discord, Ollivier and Zurek Ollivier-PRL88-2001 () suggested the use of von Neumann type measurements which consist of one-dimensional projectors that sum to the identity operator. Let the projection operators describe a von Neumann measurement for subsystem only, then the conditional density operator associated with the measurement result is
where the probability equals . The quantum conditional entropy with respect to this measurement is given by Luo-PRA77-2008 ()
and the associated quantum mutual information of this measurement is defined as
The obstacle to computing quantum discord lies in this complicated maximization procedure for calculating the classical correlation because the maximization is to be done over all possible von Neumann measurements of . Once is in hand, quantum discord is simply obtained by subtracting it from the quantum mutual information,
For a general two-qubit X-state, the quantification of quantum discord is still missing with only partial results available for subsets of three parameters Luo-PRA77-2008 (); Dillenschneider-PRB78-2008 (); Sarandy-arXiv (). An extension to five real parameters was considered although applications were limited to a smaller subset Werlang-PRA-2009 (). We provide a method to compute the classical correlation and quantum discord for a general two-qubit X-state which depends on seven real parameters. This class includes the maximally entangled Bell states, ‘Werner’ states Wer-PRA89 () which include both separable and nonseparable states, as well as others. We have evaluated an analytical expression which enables us to compute the classical correlation and quantum discord in terms of the density matrix elements of a given X-state. We examine the relation between classical correlation, quantum discord, and entanglement for various initial states.
This paper is organized as follows. In Section II, we describe some basic properties of X-states and how to calculate the classical correlation and quantum discord for them. In Section III, we apply this for various examples of X-states, studying the relation between the classical correlation, quantum discord, and entanglement. We conclude our work in Section IV. An Appendix presents details while extending also the calculation from von Neumann measurements to more general positive operator valued measurements (POVM) of one subsystem.
Ii Two-qubit X-states
In this section, we limit our discussion to initially prepared arbitrary two-qubit X-states. The density matrix of a two-qubit X-state in the representation spanned by two-qubit product states , , , is of the general form
that is, . This visual appearance resembling the letter has led them to be called X-states YuEberly-2007 () but, recently, an underlying symmetry structure of these states has been examined Rau-JPA09 ().
Eq. (7) describes a quantum state provided the unit trace and positivity conditions , , and are fulfilled. X-states are entangled if and only if either or . Both conditions cannot hold simultaneously STV-PRA98 (). Eq. (7) is a -real parameter state with three real parameters along the main diagonal and two complex (or four real) parameters at off-diagonal positions.
The eigenvalues of the density matrix in Eq. (7) are given by
The quantum mutual information is given as
where and are the marginal states of , and
After computing the quantum mutual information, we need next to compute the classical correlation . We consider projective measurements for subsystem (the projective measurements for subsystem give the same results if we restrict to either , or to ). We follow the procedure of Luo-PRA77-2008 () except that we are considering a more general class of states than the three-parameter family of that study.
It is known that any von Neumann measurement for subsystem can be written as Ref. Luo-PRA77-2008 ()
where is the projector for subsystem along the computational base and is a unitary operator with unit determinant. After the measurement, the state will change to the ensemble , where
and . The , with are of subsystem A and thus density matrices.
We may write any as
with and . This implies that these parameters, three among them independent, assume their values in the interval , i. e. for . The ensemble can be characterized by their eigenvalues as per a derivation given in the Appendix. The two eigenvalues each of and are given as
The corresponding probabilities are given as
We have defined and that generalize a single expression in Ref. Luo-PRA77-2008 () as
where , and and are the real and imaginary parts of the complex number . We have defined the parameters, , , , and as
With , Eqs. (16) and (17) for a given density matrix depend on three real parameters , , and . It can be readily checked that , , and . These three parameters are a recasting of the three independent parameters in Eq. (13) and are related to the set of Ref. Luo-PRA77-2008 () through .
The entropies of the ensemble are given as
The quantum conditional entropy in Eq. (3) is given as
As per Eq. (5), the classical correlation is obtained as
Therefore, to calculate the classical correlation and consequently quantum discord, we have to minimize the quantity (Eq. (21)) with respect to the von Neumann measurements.
To minimize Eq. (21) by setting equal to zero its partial derivatives with respect to and , observe first that the expression is symmetric under the interchange of and . It is, therefore, an even function of and the extremum lies at or at the end points or . From the definition of these parameters in Eq. (18), the end points require or and, therefore, . On the other hand, for the case , we have , and and the minimization of is equal to the minimization of either or . Further, with in these expressions involving and only linearly, extreme values are attained only at their end points: and . Thus, the maximum classical correlation and thereby the quantum discord can be obtained easily analytically. The Appendix shows how we may generalize to positive operator valued measurements (POVM) to get final compact expressions that are simple extensions of the more limited von Neumann measurements, thereby yielding the same value for the maximum classical correlation and discord.
For the special case of two-qubit X-states with restrictions , , and with real off-diagonal elements, we define
and associated entropy as
In addition, , and the minimum value of is equal to the minimum value of , which is given as
Therefore, we recover the results of Ref. Luo-PRA77-2008 () as a special case of ours.
Iii Relation between discord and entanglement
In this section, we study the relation between the classical correlation, quantum discord, and entanglement for various initial states.
As a first example, we take maximally entangled pure states, that is, the four Bell states given as , and . It is well known that for any Bell state, we have
For this particular case, quantum discord and any measure of entanglement coincide and are equal to the maximum value of the correlation. Surprisingly, if we mix any two Bell states, for example, , the classical correlation is not affected at all, and quantum discord is captured by a measure of entanglement called the entanglement of formation Bennett-PRA54-1996 (); Wootters-PRL80-1998 (). The quantum mutual information is , the classical correlation is , entanglement of formation is equal to quantum discord, i. e., Vedral-et-al (). However, this example is special as for other mixed states, entanglement and quantum discord differ substantially.
We take the class of states defined as (), where , is a maximally entangled state. Based on the results of the previous section, we are now able to calculate the classical correlation and quantum discord. We have , , and . As , the quantity is given as
As , we have . We note that , i. e., .
The classical correlation is given as
The quantum mutual information is
and quantum discord
To study the relation between quantum discord and entanglement, we choose a measure of entanglement. Although various measures of entanglement Bennett-PRA54-1996 (); Wootters-PRL80-1998 (); Vidal-PRA65-2002 () give the same result for separable states and for Bell states, the amount of entanglement of a specific mixed state is different for different measures. We prefer to compare quantum discord with concurrence Wootters-PRL80-1998 (). The concurrence for this state is given as
Figure 1 displays the classical correlation, quantum discord, and concurrence for for various values of the parameter . The solid line presents quantum discord, the dotted-dashed line is for concurrence, whereas the dashed line is for classical correlation. It can be seen that for this particular initial state, quantum discord is always less than concurrence but always greater than the classical correlation.
We consider the initial state (), where , is a maximally entangled state. In this case , and . Therefore, , which is the minimum when compared with , that is, . Hence the classical correlation is given as
The quantum mutual information is given as
and quantum discord as
with as in Eq. (LABEL:Eq:min2) . The concurrence for these states is again given as in Eq. (34), .
The classical correlation, quantum discord, and concurrence have been plotted in Figure 2 for against parameter . The solid line is for quantum discord, the dotted-dashed line is for concurrence, and the dashed line for classical correlation. Interestingly, for this particular initial state, the classical correlation is always greater than both entanglement and quantum discord except for and . Moreover, quantum discord is always less than concurrence and the classical correlation.
Let us consider the Werner state Wer-PRA89 ()
where is a maximally entangled state and . The Werner state has a peculiar property that the eigenvalues of its ensemble do not depend on the parameters and , i. e. or in other words , and we have Luo-PRA77-2008 ()
and quantum discord
The concurrence for the Werner state is given by
These correlations are plotted in Figure 3. In contrast to previous examples, the correlations have a different order as functions of , with quantum discord initially larger than concurrence and the classical correlation for , but for the range , concurrence becoming larger than discord and the classical correlation. Such a behavior is different from the previous two examples.
Finally we consider the initial state
In contrast with example (), this state also contains the noisy component. We note that , , and . We find , that is, . We have
and quantum discord
The concurrence for this state is given as
We display the classical correlation, quantum discord, and concurrence versus for this state in Figure 4. We can see that all these correlations are symmetric about . As the parameter varies from to , the quantum mutual information decreases, and the classical correlation, quantum discord, and entanglement decrease, and vice versa.
We have derived analytical expressions for the classical correlation and quantum discord in X-states, a seven-parameter family of states of two qubits. This generalizes results previously available only for a three-parameter subset of such states. A large class of two-qubit states that includes maximally or partially entangled states, and mixed states that are separable or non-separable can now be examined for these various correlations. We present such results for the classical correlation, quantum discord, and entanglement for various density matrices. We conclude that there are no simple ordering relations between these correlations and, in particular, that quantum discord may be smaller or larger than entanglement as measured with concurrence or negativity. Thus, quantum discord is a fundamentally different resource than entanglement and these correlations are different qualitatively and quantitatively.
Acknowledgements.We thank Drs. Jaroslav Novotný and Joseph Rennes for discussions. MA would like to thank Prof. Christof Wunderlich for his kind hospitality at Universität Siegen where part of this work is done. MA acknowledges financial support by the Higher Education Commission, Pakistan, and by the DAAD. ARPR acknowledges support from the Alexander von Humboldt Foundation.
Appendix: Calculational details of von Neumann and POVM alternatives
where and are complex and the other coefficients real, the diagonal entries being , , , . These parameters are given in terms of the density matrix elements by , , , , and . The choice of only three coefficients , all real, corresponds to the one made in Ref. Luo-PRA77-2008 ().
Upon taking the trace of Eq. (49), we get the probabilities
in agreement with Eq. (15) upon writing in terms of and .
The density matrices themselves for subsystem A that follow from Eq. (49) are, therefore,
Their eigenvalues are the ones in Eq. (14). Note that the denominators in Eq. (17) are , while the numerators under the square root are the sum of the squares of and as follows from the properties of the Pauli matrices. The expressions for and differ in a change in sign in in Eq. (LABEL:Eq:Arho) and Eq. (50). This close connection except for a change in sign of traces back to the similar change in sign of the two von Neumann projectors in Eq. (11), . Indeed, the reduction of the lengthy calculation to this compact statement of passing from the projectors to the expressions in Eq. (LABEL:Eq:Arho), with the measurement directions replaced by the dot product of with and corresponding coefficients of the density matrix, will be useful in generalizing to other types of measurements below.
Instead of von Neumann projectors, consider more general positive operator valued measurements (POVM). For instance, choose three orthogonal unit vectors mutually at 120,
and corresponding projectors
and of Eq. (LABEL:Eq:Arho) for with exactly similar expressions except replace in Eq. (LABEL:Eq:Arho) and Eq. (50). Here, we have defined two counterparts of that follow from the other two projectors in Eq. (54),
Other choices of projectors may define a similar third set of parameters in Ref. Luo-PRA77-2008 (), all these unit vectors being mutually orthogonal.
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