Geometric picture of quantum discord for two-qubit quantum states

Abstract

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 analytical 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.

Quantum discord of two-qubit rank-2 states

“…We will illustrate this result in the geometric picture presented in [26]. In this subsection we discuss the case of entangled states.…”

“…In calculating S (2) min , we adopt the approach provided in [40]. That is, the optimization only concerns the equi-entropy decomposition and quasi-eigendecomposition of ρ B [26]. It should be mentioned that this approach is not strictly correct [41][42][43].…”

Quantum correlations of two-qubit states with one maximally mixed marginal

“…Quantum steering ellipsoids provide a faithful and intuitive representation of two-qubit states [1][2][3][4]. If Alice and Bob each hold a qubit of a non-product state then Alice's Bloch vector is 'steered' when Bob performs a local measurement.…”

An easy measure of quantum correlation

“…Most of the above measures coincide with the quantum entanglement of relative entropy for bipartite pure states, and none of them coincides with the widely used entanglement measure, Wootters concurrence [28]. Furthermore, to satisfy the second basic condition for a good measure mentioned above, an optimization is required in the definition of the above measures, which makes it very hard to analytically evaluate the quantum correlation of a mixed state [29,30], even for the simplest two-qubit system [31][32][33]. A related discussion on the optimization over positive-operator-valued measurements can be found in Ref.…”