Abstract
We present an analytical solution for classical correlation, defined in terms of linear entropy, in an arbitrary 
system when the second subsystem is measured. We show that the optimal measurements used in the maximization of the classical correlation in terms of linear entropy, when used to calculate the quantum discord in terms of von Neumann entropy, result in a tight upper bound for arbitrary 
systems. This bound agrees with all known analytical results about quantum discord in terms of von Neumann entropy and, when comparing it with the numerical results for 106 two-qubit random density matrices, we obtain an average deviation of order 10−4. Furthermore, our results give a way to calculate the quantum discord for arbitrary n-qubit GHZ and W states evolving under the action of the amplitude damping noisy channel.
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Introduction
Quantum entanglement plays important roles in many areas of quantum information processing, such as quantum teleportation and superdense coding1,2,3. Nevertheless, quantum entanglement is not the only form of quantum correlation that is useful for quantum information processing. Indeed, some separable states may also speed up certain quantum tasks, relative to their classical counterparts4,5,6,7 and many quantum tasks, such as quantum nonlocality2,8,9 and deterministic quantum computations with one qubit10, can be carried out with forms of quantum correlation other than quantum entanglement. One such quantum correlation, called quantum discord, has received a great deal of attention recently (see ref. 11 and references therein). Introduced by Ollivier and Zurek12 as the difference between the quantum mutual information and the maximal conditional mutual information obtained by local measurements12,13, quantum discord plays an important role in some quantum information processing14,15. Despite much effort by the scientific community, an analytical solution of quantum discord is still lacking even for two-qubit systems. Owing to the maximization involved in the calculation, there are only a few results on the analytical expression of quantum discord and only for very special states are exact solutions known. However, if instead of the von Neumann entropy one uses the linear entropy, the optimal measurements that maximize the conditional mutual information can be obtained analytically16. Here, we show that using these optimal measurements to determine the quantum discord in terms of the von Neumann entropy results in an excellent upper bound for the latter. Moreover, we show that this result gives a way to calculate the quantum discord for arbitrary n-qubit GHZ and W states, with each qubit subjected to the amplitude damping channel individually.
Results
Classical correlation under linear entropy
The usual quantum discord, in terms of von Neumann entropy, is defined as follows: let 
denote the density operator of a bipartite system composed of partitions A and B. Let 
and 
be the reduced density operators of A and B, respectively. The quantum mutual information, which is the information-theoretic measure of the total correlation, is defined as 
, where 
is the von Neumann entropy. Usually, the total correlation 
is split into the quantum part 
and the classical part 
, such that 
. The classical correlation of a bipartite state 
is defined as
where the maximum is taken over all positive operator-valued measurements (POVM) 
performed on subsystem B, satisfying 
, with probability of i as an outcome, 
where 
is the conditional state of system A associated with outcome i, where 
is the identity operator on subsystem A.
In this work, all POVM or projective measurements (PM) are taken on subsystem B. Finally, the quantum discord is defined as the difference between the total correlation and the classical correlation12,13:
where 
is the conditional entropy.
To calculate our tight upper bound to quantum discord, instead of the von Neumann entropy one uses the linear entropy. The linear entropy of a state 
is given by:
In terms of the linear entropy (3), one can correspondingly define the conditional linear entropy, 
and the classical correlation16 is written as:
where the measurements run over all POVMs Pi.
Although the classical correlation and, consequently, the quantum discord (2) is extremely difficult to compute in terms of von Neumann entropy, the classical correlation (4) expressed in terms of linear entropy can be calculated analytically. In what follows we present the analytical formula for an arbitrary 
quantum systems.
A qudit state can be written as 
, where 
denotes the 
identity matrix, 
is a 
-dimensional real vector, 
is the vector of generators of 
and 
stands for transpose. Consider a bipartite system, composed of a 
-dimensional subsystem labeled A and a 2-dimensional subsystem labeled B. The bipartite state 
can be written as:
where 
is the symmetric two-qubit purification of the reduced density operator 
on an auxiliary qubit system 
and 1 is the identity map on system B. Here, symmetric two-qubit purification means that the two reduced density matrices are equal, i.e. 
and 
is a a completely positive trace-preserving map which maps a qubit state 
to the qudit state A. Let 
denote the vector of Pauli operators, r being a three-dimensional vector, 
. As a qubit state can generally be written as 
, the map 
is of the form
where L is a 
real matrix and s is a three-dimensional vector. L and s can be obtained from Eq. (5) and Eq. (6). Let 
be the spectral decomposition of 
. Then 
and 
, 
, can be calculated by Eq. (5). Therefore one gets 
, 
and the matrix 
. By the method used to calculate the classical correlation 
of two-qubit states16, we have:
where 
stands for the largest eigenvalue of the matrix 
. Eq. (7) gives the analytical formula for the classical correlation in terms of linear entropy for a general 
quantum state. Indeed, one only needs to find the eigenvalues of the matrix 
.
Since, for a given state 
, the reduced state 
, 
and the map 
are fixed, the classical correlation can readily be computed in terms of linear entropy 
. What concern us here are the optimal measurements that give rise to 
. In fact, there is a one-to-one correspondence between all possible POVM measurements and all convex decompositions of 
Ref. 17; namely, if 
is the pure state decomposition of 
, then the following are the corresponding POVMs:
where 
is full-ranked. Otherwise, we can find the inverse of 
in its range projection and, from the optimal pure state decompositions of 
, we can get the corresponding optimal POVMs. In Ref. 16, the authors have shown how to find the optimal decomposition of 
. First write 
in its Bloch form: 
. Let 
be the Bloch vector for the pure state decomposition 
of 
, where 
and 
, 
. Hence, 
.Then 
. Without loss of generality, assume that 
is diagonal with diagonal elements 
Eq. (7) becomes 
, which gets the maximum value when 
. There are exactly two solutions of the equation 
. Hence the optimal decomposition of 
reads: 
. From the two pure states in the optimal decomposition, we obtain the two optimal POVM measurement operators 
and 
.
It is well known that to maximize the classical correlation it is necessary to use the most general POVM quantum measurement. As it is much more complicated to find the maximum in (1) over all POVMs than over von Neumann measurements, almost all known analytical results are based on the latter. Indeed, only very few results are based on POVM18,19. Here, we show that for the case of a bipartite qudit-qubit state, the classical correlation based on linear entropy is maximized over projective measurements (see proof in the appendix). This leads to our first theorem:
Theorem 1. The classical correlation of a qudit-qubit state 
defined by running over all (arbitrary) POVM measurements is the same as the classical correlation defined by running over all projective measurements, i.e., 
.
Quantum discord under von Neumann entropy
Theorem 1 implies that the optimal POVM in the classical correlation defined by Eq. (4) is in fact a projective measurement. This is very different from the case of classical correlation 
defined by von Neumann entropy, in which the classical correlation based on POVM could be larger than the one based on projective measurement18,19. This shows that, although von Neumann entropy and linear entropy have many properties in common, they behave quite differently in optimizing classical information. However, by using the optimal projective measurement for the classical correlation 
based on linear entropy, we can get a tight lower bound for the classical correlation based on von Neumann entropy and hence a tight upper bound for the quantum discord based on von Neumann entropy. This leads us to our second theorem:
Theorem 2. The quantum discord based on von Neumann entropy has an upper bound:
where 
is the probability of the measurement outcome 
, 
is the conditional state of system A when the measurement outcome is i and 
and 
are the optimal projective measurement operators for 
of a given 
state 
.
In fact, there is a connection between discord and entanglement of formation (EOF): the classical correlation 
can be obtained from EOF by the Koashi Winter Relation20,
where 
is the original classical correlation of 
state 
, E(ρAC ) is the EOF of state 
and 
is the purification of 
. It is important to note that, from theorem 2, we can get an upper bound of EOF for arbitrary rank two 
state 
.
Although the upper bound (10) of 
is given by the optimal measurement of 
, we show, by means of examples, that it is a surprisingly good estimate of 
.
Example 1. In Ref. 21 Luo presented the analytic formula for the quantum discord 
of the two-qubit Bell-diagonal state: 
. Let 
. For this Bell-diagonal state, 
and 
. The two solutions of 
are 
and 
when 
: 
and 
; 
and 
when 
: 
and 
; 
and 
when 
: 
and 
. It can be verified immediately that the optimal measurements for 
are given by 
and 
, for 
with 
. It can easily be checked that our upper bound (10) is exactly the same as the analytical results in ref. 21.
Example 2. In ref. 22,23 the X-type two-qubit states are investigated: 
, where 
, 
, 
, 
and 
are defined such that 
is a quantum state. It can easily be seen that our upper bound (10) agrees perfectly with the analytical results obtained in ref. 22 (see Fig. 1).
Quantum discord 
for 
, 
, 
, 
. Here the results in 24, our numerical results and our upper bound in Eq. (10) agree with high precision.
Now, let us consider the following general two-qubit states, 
and compare our analytical upper bound with numerical results. Figure 2 gives the quantum discord 
, for 
, 
, 
, 
, 
, 
, 
and 
plotted against 
, such that 
is a quantum state. Figure 3 shows the quantum discord 
for 
, 
, 
, 
, 
, 
, 
and 
, plotted against 
, such that 
is a quantum state. It can be seen that our upper bound coincides very well with the numerical results.
Figure (a) shows quantum discord 
. Solid blue line shows numerical results and the red dotted line our upper bound. Figure (b) shows the difference between the numerical results and our upper bound.
Figure (a) shows quantum discord 
. Solid blue line shows numerical results and the red dotted line our upper bound. Figure (b) shows the difference between the numerical results and our upper bound.
We have seen that the upper bound of quantum discord based on von Neumann entropy, obtained from the optimal measurements for the classical correlation based on linear entropy, is often exact. To test the precision of our upper bound generally, we calculated the difference between our analytical result and the numerical solution of quantum discord, for a set of 
random density matrices of 
. In Fig. 4, we plot the deviation 
against the number of occurrences. It can be seen that more than half of the randomly generated density matrices results in a precision greater than 
, which demonstrates that our analytical result is a tight upper bound. Furthermore, in Fig. 4, we show that more than 
of the density matrices randomly generated lead to a precision greater than 
. Indeed, the percentage of density matrices with a deviation 
greater than 
is less than 
. Here, in the horizontal coordinate of Fig. 4 
represents the interval from 0 to 1 and the same for 
, etc..
as a function of number of occurrences for a set of 
random 
density matrices.
Evolution of Quantum Discord under AD Channel
Now we consider the evolution of quantum discord for arbitrary 
-qubit GHZ and W states under an amplitude damping (AD) channel characterized by the Kraus operators 
and 
. We show that the related quantum discord based on von Neumann entropy can be analytically obtained from the upper bound given by Eq. (10).
First let us consider 
-qubit GHZ states, with the first 
qubits subjected to AD channels individually. From Theorem 2, we get the optimal measurement operators 
and 
for classical correlation in terms of linear entropy and the upper bound of quantum discord in terms of von Neumann entropy is then exact. Let 
and 
be the two measurement operators, where 
and 
are the projective operators, 
with 
and 
. Figure 5 shows that when 
or 
has the minimal value, which coincides with the optimal measurement operators 
and 
for classical correlation based on linear entropy.
as a function of 
and p.
For 
-qubit W states with the first 
qubits subjected to individual AD channels, from Theorem 2 we have the optimal measurement operators 
and 
or 
and 
. The upper bound of quantum discord obtained in terms of these measurement operators coincide with its lower bound in ref. 24. It follows that again we have the exact value of quantum discord (2).
Alternatively, if the last qubit of an 
-qubit W state is subjected to an AD channel, we have the optimal measurement operators 
and 
or 
and 
, which also give rise to the exact value of discord (2).
Conclusions
We have studied the quantum discord of qudit-qubit states. The analytical formula for classical correlation based on linear entropy has been explicitly derived, from which an analytical tight upper bound of quantum discord based on von Neumann entropy is obtained for arbitrary qudit-qubit states. The upper bound is found to be surprisingly good in the sense that it agrees very well with all known analytical results about quantum discord in terms of von Neumann entropy. Furthermore, for a set of 
random density matrices, the maximum deviation found from the numerical solution was approximately 
and the number of density matrices whose deviation was greater than 
was less than 
of the whole set. Our analytical results could be used to investigate the roles played by quantum discord in quantum information processing. For classical correlation in terms of linear entropy, it has also been shown that the result for a qudit-qubit state, defined by running over all two-operator POVM measurements, is equivalent to that defined by running over all projective measurements. Furthermore, our results can be applied to investigate the evolution of quantum discord for arbitrary 
-qubit GHZ and W states. Indeed, employing an important paradigmatic noisy channel, we present the quantum discord dynamics for the GHZ and W states when each qubit is subjected to independent amplitude damping channels.
Additional Information
How to cite this article: Ma, Z. et al. Quantum Discord for d⊗2 Systems. Sci. Rep. 5, 10262; doi: 10.1038/srep10262 (2015).
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Acknowledgements
The work is supported by NSFC under numbers 11371247, 10901103 and 11201427. FFF is supported by São Paulo Research Foundation (FAPESP), under grant number 2012/50464-0 and by the National Institute for Science and Technology of Quantum Information (INCT-IQ), under process number 2008/57856-6. FFF is also supported by the National Counsel of Technological and Scientific Development (CNPq) under grant number 474592/2013-8.
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Contributions
Z.M. and S.F. prove the main theorems, Z.C. and F.F.F. developed the numerical codes and Z.M., Z.C., F.F.F. and S.F. wrote the manuscript.
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Ma, Z., Chen, Z., Fanchini, F. et al. Quantum Discord for d⊗2 Systems. Sci Rep 5, 10262 (2015). https://doi.org/10.1038/srep10262
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