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 Neuman
n 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.
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.
