Unified View of Quantum and Classical Correlations

Abstract

We discuss the problem of separation of total correlations in a given quantum state into entanglement, dissonance, and classical correlations using the concept of relative entropy as a distance measure of correlations. This allows us to put all correlations on an equal footing. Entanglement and dissonance, whose definition is introduced here, jointly belong to what is known as quantum discord. Our methods are completely applicable for multipartite systems of arbitrary dimensions. We investigate additivity relations between different correlations and show that dissonance may be present in pure multipartite states.Introduction.-Quantum systems are correlated in ways inaccessible to classical objects. A distinctive quantum feature of correlations is quantum entanglement [1][2][3]. Entangled states are nonclassical as they cannot be prepared with the help of local operations and classical communication (LOCC) [4]. However, it is not the only aspect of nonclassicality of correlations due to the nature of operations allowed in the framework of LOCC. To illustrate this, one can compare a classical bit with a quantum bit; in the case of full knowledge about a classical bit, it is completely described by one of two locally distinguishable states, and the only allowed operations on the classical bit are to keep its value or flip it. To the contrary, quantum operations can prepare quantum states that are indistinguishable for a given measurement. Such operations and classical communication can lead to separable states (those which can be prepared via LOCC) which are mixtures of locally indistinguishable states. These states are nonclassical in the sense that they cannot be prepared using classical operations on classical bits.Recent measures of nonclassical correlations are motivated by different notions of classicality and operational means to quantify nonclassicality [5][6][7][8][9]. Quantum discord has received much attention in studies involving thermodynamics and correlations [10][11][12], positivity of dynamics [13,14], quantum computation [15][16][17][18], broadcasting of quantum states [19,20], dynamics of discord [21][22][23], and volume of discord [24,25]. Most of these works are limited to studies of bipartite correlations only as the concept of discord, which relies on the definition of mutual information, is not defined for multipartite systems. In some of the studies, it is also desirable to compare various notions of quantum correlations. It is well known that the different measures of quantum correlation are not identical and conceptually different. For example, the discord does not coincide with entanglement and a direct comparison of two notions is rather meaningless. Therefore, an unified classification of correlations is in demand.In this Letter, we resolve these two issues by introducing a measure for classical and nonclassical correlations

Multipartite quantum and classical correlations in symmetric n-qubit mixed states

“…[11]). According to this criterion, genuine total correlations T (n) ( ) coincide with the distance, measured through the relative entropy, between and the closest state with no n-partite correlations, that is, the closest state which is product at least along a bipartite cut [13]. This definition implies that T (n) coincides with the minimum bipartite mutual information present in the system.…”

“…A series of postulates any good measure of multipartite correlations should obey was given by Bennett et al [11]. Three main generalizations of quantum discord to the multipartite case have been proposed: Rulli and Sarandy introduced the so-called global discord (GD) which is a natural extension of a symmetrized version of the QD and is based on a collective measurement [12], Modi and co-workers proposed a unified view of correlations that applies in both the bipartite and the multipartite scenarios based on the use of the relative entropy to quantify the 'distance' between states [13], and in Ref. [14], relative entropy was also employed to give a measure of genuine total, classical, and quantum correlations.…”

Unified scheme for correlations using linear relative entropy

“…The concept of relative entropy provides a unified approach to treat equitably the different kinds of correlations existing in multipartite systems [40]. In this approach, total, quantum and classical correlations satisfy a closed additivity relation.…”

“…The relative entropy offers the appropriate scheme to unify the different kinds of correlations existing in multipartite systems [40]. It is the quantum analogue of the Kullback-Leibler divergence between two classical probability distributions and characterizes the dissimilarity between two quantum states.…”

“…This classification requires a specific measure (entropic or geometric distance) to decide about the dissimilarity between a given quantum state and its closest one without the desired property and to provide a consistent scheme to treat equally the different correlations. In this sense, using the relative entropy, an approach unifying the correlations in multipartite systems was recently developed in [40]. In particular, a very significant and interesting additivity relation was reported (D + C = T + L).…”