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arXiv:1202.1959 · Physical Review A

Correlations in quantum states and the local creation of quantum discord

Manuel Gessner, Elsi-Mari Laine, Heinz‐Peter Breuer, Jyrki Piilo

Published May 23, 2012• 49 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physreva.85.052122

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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 correlations in a physical system are usually studied with respect to a unique and fixed decomposition of the system into subsystems, without fully exploiting the rich structure of the state space. Here, we show several examples in which the consideration of different ways to decompose a physical system enhances the quantum resources and accounts for a more flexible definition of quantumness measures. Furthermore, we give a different perspective regarding how to reassess the fact that local operations play a key role in general quantumness measures that go beyond entanglement---as discordlike ones. We propose a family of measures to quantify the maximum quantumness of a given state. For the discord-based case, we present some analytical results for $2\ifmmode\times\else\texttimes\fi{}d$-dimensional states. Applying our definition to low-dimensional bipartite states, we show that different behaviors can be reported for separable and entangled states vis-\`a-vis those corresponding to the usual measures of quantum correlations. We show that there is a close link between our proposal and the criterion to witness quantum correlations based on the rank of the correlation matrix, proposed by Daki\ifmmode \acute{c}\else \'{c}\fi{}, Vedral, and Brukner [Phys. Rev. Lett. 105, 190502 (2010)].
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.
Quantum discord provides a promising measure for the quantum correlation beyond entanglement, and may be responsible for the quantum speedup in certain quantum computing tasks. Here, we propose a necessary and sufficient condition for nonzero quantum discord in continuous variable systems, which is simple and easy to perform in terms of a marker QQQ. In order to obtain this condition, we introduce the continuous-variable local orthogonal basis of the operator space generalized from the orthogonal basis sets in local operator space for discrete variables. Based on this, we derive an explicit expression of the marker QQQ for all the two-mode states, and present examples of all Gaussian states and several non-Gaussian states. Our result provides a candidate for quantum correlation measures, and can be easily measured by a designed quantum circuit.
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 propose a measure of nonclassical correlations in bipartite quantum states based on local unitary operations. We prove that the measure is nonzero if and only if the quantum discord is nonzero; this is achieved via a new characterization of zero discord states in terms of the state's correlation matrix. Moreover, our scheme can be extended to ensure that the same relationship holds even with a generalized version of quantum discord in which higher-rank projective measurements are allowed. We next derive a closed-form expression for our scheme in the cases of Werner states and $(2\ifmmode\times\else\texttimes\fi{}N)$-dimensional systems. The latter reveals that for $(2\ifmmode\times\else\texttimes\fi{}N)$-dimensional states, our measure reduces to the geometric discord [Daki\ifmmode \acute{c}\else \'{c}\fi{} et al., Phys. Rev. Lett. 105, 190502 (2010)]. A connection to the Clauser-Horne-Shimony-Holt inequality is shown. We close with a characterization of all maximally nonclassical, yet separable, $(2\ifmmode\times\else\texttimes\fi{}N)$-dimensional states of rank at most 2 (with respect to our measure).