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arXiv:1501.06855 · Physical Review Letters

Hierarchy of Efficiently Computable and Faithful Lower Bounds to Quantum Discord

Marco Piani

Published Aug 15, 2016• 14 citations
arXivHTMLPDF

Abstract

Quantum discord expresses a fundamental nonclassicality of correlations that is more general than entanglement, but that, in its standard definition, is not easily evaluated. We derive a hierarchy of computationally efficient lower bounds to the standard quantum discord. Every nontrivial element of the hierarchy constitutes by itself a valid discordlike measure, based on a fundamental feature of quantum correlations: their lack of shareability. Our approach emphasizes how the difference between entanglement and discord depends on whether shareability is intended as a static property or as a dynamical process.

DOI
10.1103/physrevlett.117.080401

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Quantum discord expresses a fundamental nonclassicality of correlations that is more general than entanglement, but that, in its standard definition, is not easily evaluated. We derive a hierarchy of computationally efficient lower bounds to the standard quantum discord. Every nontrivial element of the hierarchy constitutes by itself a valid discordlike measure, based on a fundamental feature of quantum correlations: their lack of shareability. Our approach emphasizes how the difference between entanglement and discord depends on whether shareability is intended as a static property or as a dynamical process.
Quantum discord quantifies quantum correlations beyond entanglement and assumes nonzero values, which are notoriously hard to compute, for almost all quantum states. Here we provide computable tight bounds for the quantum discord for qubit-qudit states. In the case of two qubits our lower and upper bounds coincide for a 7-parameter family of filtered XXX-states, whose quantum discords can therefore be evaluated analytically. An application to the accessible information of the binary qubit channel is also presented. For the qubit-qudit state output by the circuit of deterministic computation with one qubit, nontrivial lower and upper bounds that respect the zero-discord conditions are obtained for its quantum discord.
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Quantum discord has been studied extensively as a measure of nonclassical correlations which includes entanglement as a subset. Although it is well known that nonzero discord can exist without entanglement, the origin of quantum discord is not well understood as compared to entanglement, which manifests itself more simply as inseparable higher-dimensional quantum superposition. In this paper we establish the discordlike correlation of bipartite coherence and then compare it to quantum discord. Consequently, we show that the minimum of the discordlike correlation of coherence coincides with the original quantum discord. This demonstrates quantum discord as the irreducible correlated bipartite coherence. In addition, the discordlike correlated coherence is shown to admit the postulates of the quantum resource theory (QRT), although the original quantum discord is not a ``good'' candidate under the QRT. We also find that the relative entropy measure induced from the discordlike coherence is a well-defined coherence measure for bipartite states.
Composite quantum systems can be in generic states characterized not only by entanglement but also by more general quantum correlations. The interplay between these two signatures of nonclassicality is still not completely understood. In this work we investigate this issue, focusing on computable and observable measures of such correlations: entanglement is quantified by the negativity N\mathcal{N}N, while general quantum correlations are measured by the (normalized) geometric quantum discord DG{D}_{G}DG​. For two-qubit systems, we find that the geometric discord reduces to the squared negativity on pure states, while the relationship ${D}_{G}\ensuremath{\ge}{\mathcal{N}}^{2}$ holds for arbitrary mixed states. The latter result is rigorously extended to pure, Werner, and isotropic states of two-qudit systems for arbitrary ddd, and numerical evidence of its validity for arbitrary states of a qubit and a qutrit is provided as well. Our results establish an interesting hierarchy, which we conjecture to be universal, between two relevant and experimentally friendly nonclassicality indicators. This ties in with the intuition that general quantum correlations should at least contain and in general exceed entanglement on mixed states of composite quantum systems.
Measurements of quantum systems disturb their states. To quantify this nonclassical characteristic, Zurek and Ollivier [Phys. Rev. Lett. 88, 017901 (2001)] introduced the quantum discord, a quantum correlation that can be nonzero even when entanglement in the system is zero. Discord has aroused great interest as a resource that is more robust against the effects of decoherence and offers the exponential speed-up of certain computational algorithms. Here, we study general two-level bipartite systems and give general results on the relationship between discord, entanglement, and linear entropy. We also identify the states for which discord takes a maximal value for a given entropy or entanglement, thus placing strong bounds on entanglement-discord and entropy-discord relations. We find out that although discord and entanglement are identical for pure states, they differ when generalized to mixed states as a result of the difference in the method of generalization.
Quantum discord witnesses the nonclassicality of quantum states even when there is no entanglement in these quantum states. This type of quantum correlation also has some interesting and significant applications in quantum information processing. Quantum discord has been evaluated explicitly only for a certain class of two-qubit states. We extend the previous studies to 2⊗ d quantum systems and derive analytical expressions for quantum discord, classical correlation and entanglement for a two-parameter class of states for d ⩾ 3. We compare quantum discord, classical correlation and entanglement for qubit–qutrit systems to demonstrate that different measures of quantum correlation are not identical and conceptually different.