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Register a new userGeometric picture of quantum discord for two-qubit quantum states
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Among various definitions of quantum correlations, quantum discord has attracted considerable attention. To find analytical expression of quantum discord is an intractable task. Exact results are known only for very special states, namely, two-qubit X-shaped states. We present in this paper a geometric viewpoint, from which two-qubit quantum discord can be described clearly. The known results about X state discord are restated in the directly perceivable geometric language. As a consequence, the dynamics of classical correlations and quantum discord for an X state in the presence of decoherence is endowed with geometric interpretation. More importantly, we extend the geometric method to the case of more general states, for which numerical as well as analytica results about quantum discord have not been found yet. Based on the support of numerical computations, some conjectures are proposed to help us establish geometric picture. We find that the geometric picture for these states has intimate relationship with that for X states. Thereby in some cases analytical expressions of classical correlations and quantum discord can be obtained.
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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.
We present a dynamic approach to study the quantum discord and classical correlation. By local filtering operation, the evaluation of quantum discord is closely related to quantum channel and channel capacity. As a consequence, the traditional optimization over horizontal-or-vertical von Neumann measurements is replaced by that over horizontal-and-vertical three-element POVM measurement, from which more rigorous results of quantum discord are obtained.
We investigate the geometric picture of the level surfaces of quantum entanglement and geometric measure of quantum discord (GMQD) of a class of X-states, respectively. This pictorial approach provides us a direct understanding of the structure of entanglement and GMQD. The dynamic evolution of GMQD under two typical kinds of quantum decoherence channels is also investigated. It is shown that there exists a class of initial states for which the GMQD is not destroyed by decoherence in a finite time interval. Furthermore, we establish a factorization law between the initial and final GMQD, which allows us to infer the evolution of entanglement under the influences of the environment.
We present an efficient method to solve the quantum discord of two-qubit X states exactly. A geometric picture is used to clarify whether and when the general POVM measurement is superior to von Neumann measurement. We show that either the von Neumann measurement or the three-element POVM measurement is optimal, and more interestingly, in the latter case the components of the postmeasurement ensemble are invariant for a class of states.
The minimal Bures distance of a quantum state of a bipartite system AB to the set of classical states for subsystem A defines a geometric measure of quantum discord. When A is a qubit, we show that this geometric quantum discord is given in terms of the eigenvalues of a (2 n_B) x (2 n_B) hermitian matrix, n_B being the Hilbert space dimension of the other subsystem B. As a first application, we calculate the geometric discord for the output state of the DQC1 algorithm. We find that it takes its highest value when the unitary matrix from which the algorithm computes the trace has its eigenvalues uniformly distributed on the unit circle modulo a symmetry with respect to the origin. As a second application, we derive an explicit formula for the geometric discord of two-qubit states with maximally mixed marginals and compare it with other measures of quantum correlations. We also determine the closest classical states to such two-qubit states.
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