Abstract

Quantum correlations may be measured by means of the distance of the state to the subclass ofstates having well defined classical properties. In particular, a geometric measure of asymmetricdiscord [Daki´c et al., Phys. Rev. Lett. 105, 190502 (2010)] was recently defined as the Hilbert-Schmidt distance of a given two-qubit state to the closest classical-quantum (CQ) correlated state.We analyze a geometric measure of symmetric discord defined as the Hilbert-Schmidt distance ofa given state to the closest classical-classical (CC) correlated state. The optimal member of isjust specially measured original state both for the CQ and CC discords. This implies that thismeasure is equal to quantum deficit of post-measurement purity. We discuss some general relationsbetween the CC discords and explain why an analytical formula for the CC discord, contrary to theCQ discord, can hardly be found even for a general two-qubit state. Instead of such exact formula,we find simple analytical measurement-based upper bounds for the CC discord which, as we show,are very efficient in the case of two qubits and may serve as independent indicators of two-partyquantum correlations. In particular, we propose an adaptive upper bound, which corresponds to theoptimal states induced by single-party measurements: optimal measurement on one of the partiesdetermines an optimal measurement on the other party. We discuss how to refine the adaptiveupper bound by nonoptimal single-party measurements and by an iterative procedure which usuallyrapidly converges to the CC discord. We also raise the question of optimality of the symmetricmeasurements realising the CC discord on symmetric states, and give partial answer for the qubitcase.

Authors (5)