  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.
