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Communications in Theoretical Physics

Decomposition Properties of Quantum Discord*

Sunho Kim, Jun-De 俊德 Wu 武, Minhyung Cho

Published Apr 1, 2016• 0 citations
Publisher

Abstract

Abstract The quantum discord was introduced by Ollivier, Zurek, Henderson, and Vedral as an indicator of the degree of quantumness of mixed states. In this paper, we provide a decomposition condition for quantum discord. Moreover, we show that under the condition, the quantum correlations between the quantum systems can be captured completely by the entanglement measure. Finally, we present examples of our conclusions.

DOI
10.1088/0253-6102/65/4/441

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Abstract The quantum discord was introduced by Ollivier, Zurek, Henderson, and Vedral as an indicator of the degree of quantumness of mixed states. In this paper, we provide a decomposition condition for quantum discord. Moreover, we show that under the condition, the quantum correlations between the quantum systems can be captured completely by the entanglement measure. Finally, we present examples of our conclusions.
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.
Measurements of Quantum Systems disturb their states. To quantify this non-classical characteristic, Zurek and Ollivier introduced the quantum discord, a quantum correlation which 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 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, and 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 correlations in composite and separable quantum systems are characterized by non-vanishing quantum discord. We demonstrate the necessary and sufficient conditions for existence of hermitian witness operators for quantum discord, bypassing the problem of non-convexity of the set of all classical states. We propose an example of a linear witness which can detect quantum correlations in large classes of separable bipartite systems. Decomposition of the witness in terms of a small number of locally measurable operators as shown here, paves the way for experimental detection of quantum correlations beyond entanglement.
Quantum discord is usually referred to as a measure for quantum correlations. In the search of the fundamental resource to gain a quantum advantage in quantum information applications, quantum discord is considered a promising candidate. In this paper we present an alternative view on quantum correlations in terms of the rank of the correlation matrix as introduced by Daki\ifmmode \acute{c}\else \'{c}\fi{}, Vedral, and Brukner [Phys. Rev. Lett. 105, 190502 (2010)]. According to our analysis, information about the quantum discord does not necessarily determine the amount of quantum correlations but rather the quantumness of the state. Nonzero quantum discord is only a necessary but not a sufficient condition for correlations above the classically achievable limit. This becomes clear when we consider states of nonzero discord, which can be created from zero discord states only by a single local operation. We further show that the set of these states has measure zero.
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.