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arXiv:1207.1298 · Physical Review A

Witnessed entanglement and the geometric measure of quantum discord

Tiago Debarba, Thiago O. Maciel, Reinaldo O. Vianna

Published Aug 27, 2012• 90 citations
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Abstract

We establish relations between geometric quantum discord and entanglement quantifiers obtained by means of optimal witness operators. In particular, we prove a relation between negativity and geometric discord in the Hilbert-Schmidt norm, which has been conjectured before [D. Girolami and G. Adesso, Phys. Rev. A 84, 052110 (2011)]. We also show that, redefining the geometric discord with the trace norm, better bounds can be obtained. We illustrate our results numerically for Werner states and for families of bound entangled states.

DOI
10.1103/physreva.86.024302

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Comment on “Witnessed entanglement and the geometric measure of quantum discord”

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In a recent article [Phys. Rev. A 86, 024302 (2012)], the authors have derived some hierarchy relations between geometric discord and entanglement (measured by negativity and its square). We point out that these results are incorrect and give analytic counterexamples. We also briefly discuss the reason for such violations.

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We establish relations between geometric quantum discord and entanglement quantifiers obtained by means of optimal witness operators. In particular, we prove a relation between negativity and geometric discord in the Hilbert-Schmidt norm, which has been conjectured before [D. Girolami and G. Adesso, Phys. Rev. A 84, 052110 (2011)]. We also show that, redefining the geometric discord with the trace norm, better bounds can be obtained. We illustrate our results numerically for Werner states and for families of bound entangled states.
It has recently been pointed out that the geometric quantum discord, as defined by the Hilbert-Schmidt norm (2-norm), is not a good measure of quantum correlations, since it may increase under local reversible operations on the unmeasured subsystem. Here, we revisit the geometric discord by considering general Schatten ppp-norms, explicitly showing that the 1-norm is the only ppp-norm able to define a consistent quantum correlation measure. In addition, by restricting the optimization to the tetrahedron of two-qubit Bell-diagonal states, we provide an analytical expression for the 1-norm geometric discord, which turns out to be equivalent to the negativity of quantumness. We illustrate the measure by analyzing its monotonicity properties.
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 correlations may be measured by means of the distance from the state to the subclass of states $\ensuremath{\Omega}$ having well-defined classical properties. In particular, a geometric measure of asymmetric discord [Daki\ifmmode \acute{c}\else \'{c}\fi{} et al., Phys. Rev. Lett. 105, 190502 (2010)] was recently defined as the Hilbert-Schmidt distance from 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 from a given state to the closest classical-classical (CC) correlated state. The optimal member of $\ensuremath{\Omega}$ is just a specially measured original state for both the CQ and CC discords. This implies that this measure is equal to the quantum deficit of postmeasurement purity. We discuss some general relations between the CC discords and explain why an analytical formula for the CC discord, contrary to the CQ discord, can hardly be found even for a general two-qubit state. Instead of such an exact formula, we find simple analytical-measurement-based upper bounds for the CC discord which, as we show, are tight and faithful in the case of two qubits and may serve as independent indicators of two-party quantum correlations. In particular, we propose an adaptive upper bound, which corresponds to the optimal states induced by single-party measurements: optimal measurement on one of the parties determines an optimal measurement on the other party. We discuss how to refine the adaptive upper bound by nonoptimal single-party measurements and by an iterative procedure which usually rapidly converges to the CC discord. We also raise the question of optimality of the symmetric measurements realizing the CC discord on symmetric states and give a partial answer for the qubit case.
Abstract. We propose a modified metric based on the Hilbert-Schmidt norm and adopt it to define a rescaled version of the geometric measure of quantum discord. Such a measure is found not to suffer from the pathological dependence on state purity. Although the employed metric is still noncontractive under quantum operations, we show that the resulting indicator of quantum correlations is in agreement with other bona fide discord measures in a number of physical examples. We present a critical assessment of the requirements of reliability versus computability when approaching the task of quantifying, or measuring, general quantum correlations in a bipartite state. 1.