Quantum Discord for d⊗2 Systems

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.

Quantum entanglement plays important roles in many areas of quantum information processing, such as quantum teleportation and superdense coding [1][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 counterparts [4][5][6][7] , and many quantum tasks, such as quantum nonlocality 2,8,9 and deterministic quantum computations with one qubit 10 , 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 Zurek 12 as the difference between the quantum mutual information and the maximal conditional mutual information obtained by local measurements 12,13 , quantum discord plays an important role in some quantum information processing 14,15 . 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 analytically 16 . 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 ρ AB denote the density operator of a bipartite system composed of AB 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 Tr log 2 is the von Neumann entropy. Usually, the total correlation  is split into the quantum part  and the classical part , such that = + I Q C. The classical correlation of a bipartite state ρ AB is defined as where the maximum is taken over all positive operator-valued measurements (POVM) P i is the conditional state of system A associated with outcome i, where I A 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 correlation 12,13 : 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 ρ is given by: In terms of the linear entropy (3), one can correspondingly define the conditional linear entropy, , and the classical correlation 16 is written as: where the measurements run over all POVMs P i . 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 ⊗ d 2 quantum systems.
A qudit state can be written as ρ γ is the vector of generators of ( ) SU d and T stands for transpose. Consider a bipartite system, composed of a d-dimensional subsystem labeled A and a 2-dimensional subsystem labeled B. The bipartite state ρ AB can be written as: where ′ V B B is the symmetric two-qubit purification of the reduced density operator ρ B on an auxiliary qubit system ′ B 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 Λ is a a completely positive trace-preserving map which maps a qubit state ′ B to the qudit state A. Let σ σ σ σ = ( , , ) T 1 2 3 denote the vector of Pauli operators, r being a three-dimensional vector, ≤ r 1. As a qubit state can generally be written as ρ σ real matrix and s is a three-dimensional vector. L and s can be obtained from Eq. (5) and Eq. of two-qubit states 16 , we have: where λ ( ) L L max T stands for the largest eigenvalue of the matrix L L T . Eq. (7) gives the analytical formula for the classical correlation in terms of linear entropy for a general ⊗ d 2 quantum state. Indeed, one only needs to find the eigenvalues of the matrix L L T . Since, for a given state ρ AB , the reduced state ρ B , ′ V B B and the map Λ are fixed, the classical correlation can readily be computed in terms of linear entropy ρ ( ) ← I AB 2 . What concern us here are the optimal measurements that give rise to is the pure state decomposition of ρ B , then the following are the corresponding POVMs: where ρ B is full-ranked. Otherwise, we can find the inverse of ρ B in its range projection and, from the optimal pure state decompositions of ρ B , we can get the corresponding optimal POVMs. In Ref. 16, the authors have shown how to find the optimal decomposition of ρ B . First write ρ B in its Bloch form: . Without loss of generality, assume that L L T is diago- , which gets the maximum value when There are exactly two solutions of the equation . Hence the optimal decomposition of ρ B reads: . From the two pure states in the optimal decomposition, we obtain the two optimal POVM measurement operators M 0 and M 1 .
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 POVM 18,19 . 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 ρ AB 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 ρ ( ) ← I AB defined by von Neumann entropy, in which the classical correlation based on POVM could be larger than the one based on projective measurement 18,19 . 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 ρ ( ) ← I AB 2 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: i is the conditional state of system A when the measurement outcome is i, and P 0 and P 1 are the optimal projective measurement operators for  (10) , where x 3 , y 3 , t 1 , t 2 and t 3 are defined such that ρ 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). 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 ρ ( ) ×  2 2 , for = . , plotted against x 1 , such that ρ × 2 2 is a quantum state. It can be seen that our upper bound coincides very well with the numerical results.
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 10 6 random density matrices of ⊗ 2 2. In Fig. 4, we plot the deviation ∆ = −   Analytical Numerical 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 − 10 4 , which demonstrates that our analytical result is a tight upper bound. Furthermore, in Fig. 4, we show that more than % 80 of the density matrices randomly generated lead to a precision greater than − 10 3 . Indeed, the percentage of density matrices with a deviation ∆ greater than × − 6 10 3 is less than . 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 n-qubit W state is subjected to an AD channel, we have the optimal measurement operators σ ( + )/ I 2 , 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 10 6 random density matrices, the maximum deviation found from the numerical solution was approximately . 0 05 and the number of density matrices whose deviation was greater than × − 6 10 3 was less than . % 0 1 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 n-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.