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Physical Review A

Entanglement as minimal discord over state extensions

Shunlong Luo

Published Sep 29, 2016• 18 citations

Abstract

The characterization and quantification of quantum correlations, which play an instrumental role in exploring and exploiting the quantum world, have been extensively and intensively studied in the past few decades. Of special prominence and significance are the concepts of entanglement and discord, which are usually regarded as very distinctive quantum correlations, with the latter going beyond the former. In this work we establish a direct and natural link between entanglement and discord via state extensions and reveal that entanglement is actually the intrinsic discord, by which we mean that entanglement is the irreducible residue of discord viewed from ambient spaces. Our approach, taking into account the contextuality of a quantum state and being of a global nature, stands in sharp contrast to the local operations and classical communication paradigm of entanglement, which focuses on the state itself via a local approach. Furthermore, we introduce a figure of merit which, on the one hand, captures the essence of entanglement, i.e., nonlocality and quantumness of correlations, and, on the other hand, leads to a quantitative decomposition of total correlations into classical correlations, dissonance, and entanglement. This demystifies the meaning of entanglement from the perspective of quantum measurements and provides a unified framework for the interplay of various correlations in terms of quantum measurements and mutual information.

DOI: 10.1103/physreva.94.032129

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We propose a framework to characterize entanglement with quantum discord, both asymmetric and symmetric, over state extensions. In particular, we show that the minimal Bures distance of discord over state extensions is equivalent to the Bures distance of entanglement. This equivalence places quantum discord in a more primitive position than entanglement conceptually in the sense that entanglement can be interpreted as an irreducible part of discord over all state extensions. Based on this equivalence, we also offer an operational meaning of the Bures distance of entanglement by connecting it to quantum state discrimination. Moreover, for the relative entropy part, we prove that the entanglement measure introduced by Devi and Rajagopal [A. R. U. Devi and A. K. Rajagopal, Phys. Rev. Lett. 100, 140502 (2008)] is actually equivalent to the relative entropy of entanglement. We also provide several quantifications of entanglement based on discord measures.

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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.

In this didactic article we explore the concept of quantum correlations beyond entanglement. We begin by introducing and motivating the classically correlated states and then showing how to quantify the quantum correlations using an entropic approach, arriving at a well known measure called the quantum discord. Quantum correlations and discord are then operationally linked with the task of local broadcasting. We conclude by providing some alternative perspectives on quantum correlations and how to measure them.

Quantum discord, a measure of quantum correlation beyond entanglement, is initially defined as the discord between two classically equivalent while quantum discordant definitions of mutual information. In this paper, we report some new interpretations of discord which rely on the differences between measurement-induced effects on the local measured system and the whole system. Specifically, with proper quantitative definitions introduced in Buscemi et al (2008 Phys. Rev. Lett. 100 210504), we find that quantum discord can be interpreted as the differences of measurement-induced disturbance or information gain on the local measured system and on the whole system. Combined with previous similar results based on measurement-induced entanglement and decoherence, our results provide a unified view on quantum discord.

We provide a historical perspective of how the notion of correlations has evolved within quantum physics. We begin by reviewing Shannon's information theory and its first application in quantum physics, due to Everett, in explaining the information conveyed during a quantum measurement. This naturally leads us to Lindblad's information theoretic analysis of quantum measurements and his emphasis of the difference between the classical and quantum mutual information. After briefly summarizing the quantification of entanglement using these ideas, we arrive at the concept of quantum discord, which naturally captures the boundary between entanglement and classical correlations. Finally we discuss possible links between discord, which the generation of correlations in thermodynamic transformations of coupled harmonic oscillators.