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Entropy 2018, 20(7), 500; https://doi.org/10.3390/e20070500

On the Significance of the Quantum Mechanical Covariance Matrix

1
Center for Quantum Information Science and Technology & Faculty of Engineering Sciences, Ben-Gurion University of the Negev, Beersheba 8410501, Israel
2
Physics Department, Centre for Research in Photonics, University of Ottawa, Advanced Research Complex, 25 Templeton, Ottawa, K1N 6N5, Canada
*
Author to whom correspondence should be addressed.
Received: 30 May 2018 / Revised: 24 June 2018 / Accepted: 26 June 2018 / Published: 28 June 2018
(This article belongs to the Special Issue Quantum Nonlocality)
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Abstract

The characterization of quantum correlations, being stronger than classical, yet weaker than those appearing in non-signaling models, still poses many riddles. In this work, we show that the extent of binary correlations in a general class of nonlocal theories can be characterized by the existence of a certain covariance matrix. The set of quantum realizable two-point correlators in the bipartite case then arises from a subtle restriction on the structure of this general covariance matrix. We also identify a class of theories whose covariance has neither a quantum nor an “almost quantum” origin, but which nevertheless produce the accessible two-point quantum mechanical correlators. Our approach leads to richer Bell-type inequalities in which the extent of nonlocality is intimately related to a non-additive entropic measure. In particular, it suggests that the Tsallis entropy with parameter q=1/2 is a natural operational measure of non-classicality. Moreover, when generalizing this covariance matrix, we find novel characterizations of the quantum mechanical set of correlators in multipartite scenarios. All these predictions might be experimentally validated when adding weak measurements to the conventional Bell test (without adding postselection). View Full-Text
Keywords: quantum correlations; quantum bounds; nonlocality; tsallis entropy quantum correlations; quantum bounds; nonlocality; tsallis entropy
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Carmi, A.; Cohen, E. On the Significance of the Quantum Mechanical Covariance Matrix. Entropy 2018, 20, 500.

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