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arXiv:1412.0398 · International Journal of Quantum Information

Pairwise quantum and classical correlations in multi-qubits states via linear relative entropy

Mohammed Daoud A, R. Ahl Laamara, Hanane El Hadfi

Published Sep 1, 2014• 2 citations
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Abstract

The pairwise correlations in a multi-qubit state are quantified through a linear variant of relative entropy. In particular, we derive the explicit expressions of total, quantum and classical bipartite correlations. Two different bi-partioning schemes are considered. We discuss the derivation of closest product, quantum–classical and quantum–classical product states. We also investigate the additivity relation between the various pairwise correlations existing in pure and mixed states. As illustration, some special cases are examined.

DOI
10.1142/s021974991450035x

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The pairwise correlations in a multi-qubit state are quantified through a linear variant of relative entropy. In particular, we derive the explicit expressions of total, quantum and classical bipartite correlations. Two different bi-partioning schemes are considered. We discuss the derivation of closest product, quantum–classical and quantum–classical product states. We also investigate the additivity relation between the various pairwise correlations existing in pure and mixed states. As illustration, some special cases are examined.
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The distance between a quantum state and its closest state not having a certain property has been used to quantify the amount of correlations corresponding to that property. This approach allows a unified view of the various kinds of correlations present in a quantum system. In particular, using relative entropy as a distance measure, total correlations can be meaningfully separated into a quantum part and a classical part thanks to an additive relation involving only the distances between states. Here we investigate a unified view of correlations using as a distance measure the square norm, which has already been used to define the so-called geometric quantum discord. We thus also consider geometric quantifiers for total and classical correlations, finding, for a quite general class of bipartite states, their explicit expressions. We analyze the relationship among geometric total, quantum, and classical correlations, and we find that they no longer satisfy a closed additivity relation.
We derive a classification and a measure of classical- and quantum-correlation of multipartite qubit, qutrit, and in general, nnn-level systems, in terms of SU(n)(n)(n) representations of density matrices. We compare the measure for the case of bipartite correlation with concurrence and the entropy of entanglement. The characterization of correlation is in terms of the number of nonzero singular values of the correlation matrix, but that of mixed state entanglement requires additional invariant parameters in the density matrix. For the bipartite qubit case, the condition for mixed state entanglement is written explicitly in terms of the invariant paramters in the density matrix. For identical particle systems we analyze the effects of exchange symmetry on classical and quantum correlation.
We define and analyze measures of correlations for bipartite states based on trace distance. For Bell diagonal states of two qubits, in addition to the known expression for quantum correlations using this metric, we provide analytic expressions for the classical and total correlations. The ensuing hierarchy of correlations based on trace distance is compared to those based on relative entropy and Hilbert–Schmidt norm. Although some common features can be found, the trace distance measure is shown to differentiate from the others in that the closest uncorrelated state to a given bipartite quantum state is not given by the product of the marginals, and further, the total correlations are strictly smaller than the sum of the quantum and classical correlations. We compare the various correlation measures in two dynamical non-Markovian models, locally applied phase-flip channels and random external fields. It is shown that the freezing behavior, observed across all known valid measures of quantum correlations for Bell diagonal states under local phase-flip channels, occurs for a larger set of starting states for the trace distance than for the other metrics.
On the premises that total correlations in a bipartite quantum state are measured by the quantum mutual information, and that separation of total correlations into quantum and classical parts satisfies an intuitive dominance relation, we examine to what extent various entropic entanglement measures, such as the distillable entanglement, the relative entropy entanglement, the squashed entanglement, the entanglement cost, and the entanglement of formation, can be regarded as consistent measures of quantum correlations. We illustrate that the entanglement of formation often overestimates quantum correlations and thus is too big to be a genuine measure of quantum correlations. This indicates that the entanglement of formation does not quantify the quantum correlations intrinsic to a quantum state, but rather characterizes the pure entanglement needed to build the quantum state via local operations and classical communication. Furthermore, it has the consequence that, if the additive conjecture for the entanglement of formation is true (as is widely believed), then the entanglement cost, which is an operationally defined measure of entanglement with significant physical meaning, cannot be a consistent measure of quantum correlations in the sense that it may exceed total correlations. Alternatively, if the entanglement cost is dominated by total correlations, as our intuition suggests, then we can immediately disprove the additive conjecture. Both scenarios have their counterintuitive and appealing aspects, and a natural challenge arising in this context is to prove or disprove that the entanglement cost is dominated by the quantum mutual information.