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arXiv:1205.0330 · Journal of Physics A Mathematical and Theoretical

Geometric global quantum discord

Jianwei Xu

Published Sep 19, 2012• 12 citations
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Abstract

Geometric quantum discord, proposed by Dakic et al (2010 Phys. Rev. Lett. 105 190502), is an important measure for bipartite correlations. In this paper, we generalize it to multipartite states, we call the generalized version geometric global quantum discord (GGQD). We characterize GGQD in different ways, give a lower bound for GGQD and provide some special states which allow analytical GGQD.

DOI
10.1088/1751-8113/45/40/405304

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Geometric quantum discord, proposed by Dakic et al (2010 Phys. Rev. Lett. 105 190502), is an important measure for bipartite correlations. In this paper, we generalize it to multipartite states, we call the generalized version geometric global quantum discord (GGQD). We characterize GGQD in different ways, give a lower bound for GGQD and provide some special states which allow analytical GGQD.
The original definition of quantum discord of bipartite states was defined under one-side projective measurements, it describes quantum correlation more extensively than entanglement. Dakic, Vedral, and Brukner [Phys. Rev. Lett. \textbf{105}, 190502 (2010)] introduced a geometric measure of quantum discord, and Luo, Fu [Phys. Rev. A \textbf{82}, 034302 (2010)] simplified the expression of it. In this paper we generalize the quantum discord to the case of two-side projective measurements, and also define a geometric measure on it. Further, a simplified expression and a lower bound of this geometric measure are derived and explicit expressions are obtained for some special cases.
Quantum discord, as introduced by Ollivier and Zurek (2001 Phys. Rev. Lett. 88 017901), is a measure of the discrepancy between quantum versions of two classically equivalent expressions for mutual information and is found to be useful in quantification and application of quantum correlations in mixed states. It is viewed as a key resource present in certain quantum communication tasks and quantum computational models without containing much entanglement. An early step toward the quantification of quantum discord in a quantum state was by Dakic et al (2010 Phys. Rev. Lett. 105 190502) who introduced a geometric measure of quantum discord and derived an explicit formula for any two-qubit state. Recently, Luo and Fu (2010 Phys. Rev. A 82 034302) introduced a generic form of the geometric measure of quantum discord for a bipartite quantum state. We extend these results and find generic forms of the geometric measure of quantum discord and total quantum correlations in a general N -partite quantum state. Further, we obtain computable exact formulas for the geometric measure of quantum discord and total quantum correlations in an N -qubit quantum state. The exact formulas for the N -qubit quantum state can be used to get experimental estimates of the quantum discord and the total quantum correlation.
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.
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.