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 $X$ 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 $X$ states, whose quantum discord can be evaluated analytically without any minimization, for which their algorithm is valid, and also identify a family of $X$ states for which their algorithm fails. And then we demonstrate that this special family of $X$ states provides furthermore an explicit example for the inequivalence between the minimization over positive operator-valued measures and that over von Neumann measurements.

• Received 9 February 2011

DOI:https://doi.org/10.1103/PhysRevA.84.042313