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arXiv:1411.3772 · Quantum Information Processing

General bounds for quantum discord and discord distance

Feng Liu, Guojing Tian, Qiaoyan Wen, Fei Gao

Published Feb 12, 2015• 8 citations
arXivHTMLPDF
DOI
10.1007/s11128-015-0944-x

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•Oct 10, 2010•1 citation
Dakic, Vedral, and Brukner [B. Dakic, V. Vedral, and \ifmmode \check{C}\else \v{C}\fi{}. Brukner, Phys. Rev. Lett. 105, 190502 (2010)] gave a geometric measure of quantum discord in a bipartite quantum state as the distance of the state from the closest classical quantum (or zero discord) state and derived an explicit formula for a two-qubit state. Further, Luo and Fu [S. Luo and S. Fu, Phys. Rev. A 82, 034302 (2010)] obtained a generic form of this geometric measure for a general bipartite state and established a lower bound. In this Brief Report we obtain a rigorous lower bound to the geometric measure of quantum discord in a general bipartite state which dominates that obtained by Luo and Fu.
Two-qubit X-state is a large class of quantum states which plays an important role in the quantification and dynamical study of quantum correlations. However, the corresponding quantification of quantum discord is still missing for bona fide discord measures, like original quantum discord, Bures distance of discord, and relative entropy of discord. In this paper, we consider the calculation of Bures distance of discord, which is a kind of correlation satisfying all criteria of a discord measure, for two-qubit X-states. Firstly, we derive an explicit expression for Bures distance of discord for a kind of five-parameters family of states. Moreover, for general two-qubit X-states, we not only calculate the Bures distance of discord for a subset of two-qubit X-states by classifying and analyzing the optimal local measurements and the optimal projection operators but also provide an analytic upper bound for the entirety.
Dakic, Vedral and Brukner [Physical Review Letters \tf{105},190502 (2010)] gave a geometric measure of quantum discord in a bipartite quantum state as the distance of the state from the closest classical quantum (or zero discord) state and derived an explicit formula for a two qubit state. Further, S.Luo and S.Fu [Physical Review A \tf{82}, 034302 (2010)] obtained a generic form of this geometric measure for a general bipartite state and established a lower bound. In this brief report we obtain a rigorous lower bound to the geometric measure of quantum discord in a general bipartite state which dominates that obtained by S.Luo and S.Fu.