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arXiv:0707.2195 · Physical Review Letters

Generalized Information Theoretic Measure to Discern the Quantumness of Correlations

A. R. Usha Devi, A. K. Rajagopal

Published Apr 11, 2008• 51 citations
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Abstract

A novel measure, quantumness of correlations Q_AB is introduced here for bipartite states, by incorporating the required measurement scheme crucial in defining any such quantity. Quantumness coincides with the previously proposed measures in special cases and it vanishes for separable states--a feature not captured by the measures proposed earlier. It is found that an optimal generalized measurement on one of the parts leaves the overall state in its closest separable form, which shares the same marginal for the other part, implying that Q_AB is nonzero for all entangled bipartite states and it serves as an upper bound to the relative entropy of entanglement.

DOI
10.1103/physrevlett.100.140502

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A novel measure, quantumness of correlations Q_AB is introduced here for bipartite states, by incorporating the required measurement scheme crucial in defining any such quantity. Quantumness coincides with the previously proposed measures in special cases and it vanishes for separable states--a feature not captured by the measures proposed earlier. It is found that an optimal generalized measurement on one of the parts leaves the overall state in its closest separable form, which shares the same marginal for the other part, implying that Q_AB is nonzero for all entangled bipartite states and it serves as an upper bound to the relative entropy of entanglement.
We propose a general measure of nonclassical correlations for bipartite systems based on generalized entropic functions and majorization properties. Defined as the minimum information loss due to a local measurement, in the case of pure states it reduces to the generalized entanglement entropy, i.e., the generalized entropy of the reduced state. However, in the case of mixed states it can be nonzero in separable states, vanishing just for states diagonal in a general product basis, like the quantum discord. Simple quadratic measures of quantum correlations arise as a particular case of the present formalism. The minimum information loss due to a joint local measurement is also discussed. The evaluation of these measures in simple relevant cases is as well provided, together with comparison with the corresponding entanglement monotones.
A general framework is developed for separating classical and quantum correlations in a multipartite system. Entanglement is defined as the difference in the correlation information encoded by the state of a system and a suitably defined separable state with the same marginals. A generalization of the Schmidt decomposition is developed to implement the separation of correlations for any pure, multipartite state. The measure based on this decomposition is a generalization of the entanglement of formation to multipartite systems, provides an upper bound for the relative entropy of entanglement, and is directly computable on pure states. The example of pure three-qubit states is analyzed in detail, and a classification based on minimal, four-term decompositions is developed.
Generalized measurement schemes on one part of bipartite states, which would leave the set of all separable states insensitive are explored here to understand quantumness of correlations in a more general perspective. This is done by employing linear maps associated with generalized projective measurements. A generalized measurement corresponds to a quantum operation mapping a density matrix to another density matrix, preserving its positivity, hermiticity, and trace class. The positive operator valued measure (POVM) — employed earlier in the literature to optimize the measures of classical/quantum correlations — correspond to completely positive (CP) maps. The other class, the not completely positive (NCP) maps, are investigated here, in the context of measurements, for the first time. It is shown that such NCP projective maps provide a new clue to the understanding of quantumness of correlations in a general setting. Especially, the separability–classicality dichotomy gets resolved only when both the classes of projective maps (CP and NCP) are incorporated as optimizing measurements. An explicit example of a separable state — exhibiting nonzero quantum discord, when possible optimizing measurements are restricted to POVMs — is reexamined with this extended scheme incorporating NCP projective maps to elucidate the power of this approach.
We examine the evaluation of the minimum information loss due to an unread local measurement in mixed states of bipartite systems, for a general entropic form. Such a quantity provides a measure of quantum correlations, reducing for pure states to the generalized entanglement entropy, while in the case of mixed states it vanishes just for classically correlated states with respect to the measured system, as the quantum discord. General stationary conditions are provided, together with their explicit form for general two-qubit states. Closed expressions for the minimum information loss as measured by quadratic and cubic entropies are also derived for general states of two-qubit systems. As an application, we analyze the case of states with maximally mixed marginals, where a general evaluation is provided, as well as XXX states and the mixture of two aligned states.
There are two important paradigms for defining quantum correlations in quantum information theory, viz. the information-theoretic and the entanglement-separability ones. We find an analytical relation between two measures of quantum correlations, one in each paradigm, and show that only a certain cone-like region on the two-dimensional space spanned by these measures is accessible to pure three-qubit states. The information-theoretic multiparty quantum correlation measure is related to the monogamy considerations of a bipartite information-theoretic quantum correlation measure, while the entanglement-separability multiparty measure is the generalized geometric measure, a genuine multiparty entanglement measure. We also find an analytical relation between two multiparty entanglement measures, and again obtain a cone-like accessible region in this case. One of the multisite measures in this case is related to the monogamy of a bipartite entanglement measure, while the other is again the generalized geometric measure.
Quantum entanglement of pure states of a bipartite system is defined as the amount of local or marginal (i.e. referring to the subsystems) entropy. For mixed states this identification vanishes, since the global loss of information about the state makes it impossible to distinguish between quantum and classical correlations. Here we show how the joint knowledge of the global and marginal degrees of information of a quantum state, quantified by the purities or, in general, by information entropies, provides an accurate characterization of its entanglement. In particular, for Gaussian states of continuous variable systems, we classify the entanglement of two-mode states according to their degree of total and partial mixedness, comparing the different roles played by the purity and the generalized p-entropies in quantifying the mixedness and bounding the entanglement. We prove the existence of strict upper and lower bounds on the entanglement and the existence of extremally (maximally and minimally) entangled states at fixed global and marginal degrees of information. This results allow for a powerful, operative method to measure mixed-state entanglement without the full tomographic reconstruction of the state. Finally, we briefly discuss the ongoing extension of our analysis to the quantification of multipartite entanglement in highly symmetric Gaussian states of arbitrary 1 × N-mode partitions.