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.
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 [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.
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.
The concept of quantum discord aims at unveiling quantum correlations that go beyond those described by entanglement. Its original formulation [L. Henderson and V. Vedral, J. Phys. A: Math. Gen. 34, 6899 (2001); H. Ollivier and W. H. Zurek, Phys. Rev. Lett. 88, 017901 (2001)] is difficult to compute even for the simplest case of two-qubits systems. Alternative formulations have been developed to address this drawback, such as the geometric measure of quantum discord [L. Chang and S. Luo, Phys. Rev. A 87, 062303 (2013)] and the local quantum uncertainty [D. Girolami, T. Tufarelli, and G. Adesso, Phys. Rev. Lett. 110, 240402 (2013)] that can be evaluated in closed form for some quantum systems, such as two-qubit systems. We show here that these two measures of quantum discord are equivalent for $2\ifmmode\times\else\texttimes\fi{}D$ dimensional bipartite quantum systems. By considering the relevant example of N00N states for phase estimation in lossy environments, we also show that both metrics of quantum discord quantify the decrease of quantum Fisher information of the phase estimation protocol. Given their ease of computation in $2\ifmmode\times\else\texttimes\fi{}D$ bipartite systems, the geometric measure of quantum discord and the local quantum uncertainty demonstrate their relevance as computable measures of quantum discord.
Quantum discord, as introduced by Olliver and Zurek [Phys. Rev. Lett. 88, 017901 (2001)], is a measure of the discrepancy between two natural yet different quantum analogs of the classical mutual information. This notion characterizes and quantifies quantumness of correlations in bipartite states from a measurement perspective, and is fundamentally different from the various entanglement measures in the entanglement vs separability paradigm. The phenomenon of nonzero quantum discord is a manifestation of quantum correlations due to noncommutativity rather than due to entanglement, and has interesting and significant applications in revealing the advantage of certain quantum tasks. We will evaluate analytically the quantum discord for a large family of two-qubit states, and make a comparative study of the relationships between classical and quantum correlations in terms of the quantum discord. We furthermore compare the quantum discord with the entanglement of formation, and illustrate that the latter may be larger than the former, although for separable states, the entanglement of formation always vanishes and thus is less than the quantum discord.
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 geometric measures for quantum discord were recently proposed by Modi et al (2010 Phys. Rev. Lett. 104 080501) and Dakić et al (2010 Phys. Rev. Lett. 105 190502). We study the similarities and differences for total quantum correlations of Bell-diagonal states using these two geometry-based quantum discord and the original quantum discord. We show that, under non-Markovian dephasing channels, quantum discord and one of the geometric measures remain constant for a finite amount of time, but not the other geometric measure. However, all the three measures share a common sudden change point. Our study on critical point of sudden transition might be useful for keeping long-time total quantum correlations under decoherence.