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arXiv:1110.6681 · Physical Review A

Optimal measurement for quantum discord of two-qubit states

Mingjun Shi, Chunxiao Sun, Feng-Jian Jiang, Xin-Hu Yan, Jiangfeng Du

Published Jun 27, 2012• 68 citations
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Abstract

We present a complete treatment for the quantum discord of two-qubit $X$ states, by developing a geometric picture of a quantum steering ellipsoid. It is shown that either a von Neumann measurement or a three-element positive-operator-valued measurement is optimal. The condition for the latter is obtained and expressed in geometric language. We show, by using analytical as well as numerical results, that there is a systematic structure in the optimal decomposition which exists in a class of states including the $X$ states. More significantly, we establish the relation to the quantum channel by identifying the steering ellipsoid with the quantum channel ellipsoid. Thus the quantum discord and classical correlation are closely related to the concept of the entanglement entropy of the quantum channel.

DOI
10.1103/physreva.85.064104

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We present a complete treatment for the quantum discord of two-qubit XXX states, by developing a geometric picture of a quantum steering ellipsoid. It is shown that either a von Neumann measurement or a three-element positive-operator-valued measurement is optimal. The condition for the latter is obtained and expressed in geometric language. We show, by using analytical as well as numerical results, that there is a systematic structure in the optimal decomposition which exists in a class of states including the XXX states. More significantly, we establish the relation to the quantum channel by identifying the steering ellipsoid with the quantum channel ellipsoid. Thus the quantum discord and classical correlation are closely related to the concept of the entanglement entropy of the quantum channel.
Among various definitions of quantum correlations, quantum discord has attracted considerable attention. To find analytical expression of quantum discord is an intractable task. In this paper, we discuss thoroughly the case of two-qubit rank-two states. An analytical expression for the quantum discord is obtained by means of Koashi-Winter relation. A geometric picture is demonstrated by means of quantum steering ellipsoid. We point out that in this case the optimal measurement is indeed the von Neumann measurement, which is usually used in the study of quantum discord. However, for some two-qubit states with the rank larger than two, we find that three-element POVM measurement is more optimal. It means that more careful attention should be paid in the discussion of quantum discord.
Quantum discord provides a measure for quantifying quantum correlations beyond entanglement and is very hard to compute even for two-qubit states because of the minimization over all possible measurements. Recently a simple algorithm to evaluate the quantum discord for two-qubit XXX states was proposed by Ali, Rau, and Alber [Phys. Rev. A 81, 042105 (2010)] with minimization taken over only a few cases. Here we shall at first identify a class of XXX states, whose quantum discord can be evaluated analytically without any minimization, for which their algorithm is valid, and also identify a family of XXX states for which their algorithm fails. And then we demonstrate that this special family of XXX states provides furthermore an explicit example for the inequivalence between the minimization over positive operator-valued measures and that over von Neumann measurements.
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 d⊗2 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 10(6) 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.
We present a reliable algorithm to evaluate quantum discord for general two-qubit states, amending and extending an approach recently put forward for the subclass of XXX states. A closed expression for the discord of arbitrary states of two qubits cannot be obtained, as the optimization problem for the conditional entropy requires the solution to a pair of transcendental equations in the state parameters. We apply our algorithm to run a numerical comparison between quantum discord and an alternative, computable measure of nonclassical correlations, namely, the geometric discord. We identify the extremally nonclassically correlated two-qubit states according to the (normalized) geometric discord, at a fixed value of the conventional quantum discord. The latter cannot exceed the square root of the former for systems of two qubits.
We derive a general approximate solution to the problem of minimizing the conditional entropy of a qudit-qubit system resulting from a local measurement on the qubit, which is valid for general entropic forms and becomes exact in the limit of weak correlations. This entropy measures the average conditional mixedness of the postmeasurement state of the qudit, and its minimum among all local measurements represents a generalized entanglement of formation. In the case of the von Neumann entropy, it is directly related to the quantum discord. It is shown that at the lowest nontrivial order, the problem reduces to the minimization of a quadratic form determined by the correlation tensor of the system, the Bloch vector of the qubit and the local concavity of the entropy, requiring just the diagonalization of a $3\ifmmode\times\else\texttimes\fi{}3$ matrix. A simple geometrical picture in terms of an associated correlation ellipsoid is also derived, which illustrates the link between entropy optimization and correlation access and which is exact for a quadratic entropy. The approach enables a simple estimation of the quantum discord. Illustrative results for two-qubit states are discussed.
In the analysis of quantum discord, the minimization of average entropy traditionally involved over orthogonal projective measurements may be attained at more optimal decompositions by using the positive-operator-valued measure (POVM) measurements. Taking advantage of the quantum steering ellipsoid in combination with three-element POVM optimization, we show that, for a family of two-qubit X states locally interacting with Markovian non-dissipative environments, the decay rates of quantum discord show smooth dynamical evolutions without any sudden change. This is in contrast to two-element orthogonal projective measurements, in which case the sudden change of the decay rates of quantum and classical decoherences may be a common phenomenon. Notwithstanding this, we find that a subset of X states (including the Bell diagonal states) involving POVM optimization can still preserve the sudden change character as usual.