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arXiv:1912.01031 · Physical Review Research

Limitations of entropic inequalities for detecting nonclassicality in the postselected Bell causal structure

V. Vilasini, Roger Colbeck

Published Jul 17, 2020• 4 citations
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Abstract

This paper gives evidence that entropic inequalities are in general insufficient for detecting whether a distribution can be generated with classical resources in the bipartite Bell causal structure. The authors use both Shannon and Tsallis entropic inequalities as well as a general class of post-processing operations on the observed distributions to reach this conclusion.

DOI
10.1103/physrevresearch.2.033096

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This paper gives evidence that entropic inequalities are in general insufficient for detecting whether a distribution can be generated with classical resources in the bipartite Bell causal structure. The authors use both Shannon and Tsallis entropic inequalities as well as a general class of post-processing operations on the observed distributions to reach this conclusion.
Understanding cause-effect relationships is a crucial part of the scientific process. As Bell's theorem shows, within a given causal structure, classical and quantum physics impose different constraints on the correlations that are realizable, a fundamental feature that has technological applications. However, in general it is difficult to distinguish the set of classical and quantum correlations within a causal structure. Here we investigate a method to do this based on using entropy vectors for Tsallis entropies. We derive constraints on the Tsallis entropies that are implied by (conditional) independence between classical random variables and apply these to causal structures. We find that the number of independent constraints needed to characterize the causal structure is prohibitively high such that the computations required for the standard entropy vector method cannot be employed even for small causal structures. Instead, without solving the whole problem, we find new Tsallis entropic constraints for the triangle causal structure by generalizing known Shannon constraints. Our results reveal mathematical properties of classical and quantum Tsallis entropies and highlight difficulties of using Tsallis entropies for analyzing causal structures.
Tests for quantum contextuality in terms of qqq-entropies
The information-theoretic approach to Bell's theorem is developed with use of the conditional qqq-entropies. The qqq-entropic measures fulfill many similar properties to the standard Shannon entropy. In general, both the locality and noncontextuality notions are usually treated with use of the so-called marginal scenarios. These hypotheses lead to the existence of a joint probability distribution, which marginalizes to all particular ones. Assuming the existence of such a joint probability distribution, we derive the family of inequalities of Bell's type in terms of conditional qqq-entropies for all q≥1q\geq1q≥1. Quantum violations of the new inequalities are exemplified within the Clauser-Horne-Shimony-Holt (CHSH) and Klyachko-Can-Binicioǧlu-Shumovsky (KCBS) scenarios. An extension to the case of nnn-cycle scenario is briefly mentioned. The new inequalities with conditional qqq-entropies allow to expand a class of probability distributions, for which the nonlocality or contextuality can be detected within entropic formulation. The qqq-entropic inequalities can also be useful in analyzing cases with detection inefficiencies. Using two models of such a kind, we consider some potential advantages of the qqq-entropic formulation.
The information-theoretic approach to Bell's theorem is developed with use of the conditional qqq-entropies. The qqq-entropic measures fulfill many similar properties to the standard Shannon entropy. In general, both the locality and noncontextuality notions are usually treated with use of the so-called marginal scenarios. These hypotheses lead to the existence of a joint probability distribution, which marginalizes to all particular ones. Assuming the existence of such a joint probability distribution, we derive the family of inequalities of Bell's type in terms of conditional qqq-entropies for all q≥1q\geq1q≥1. Quantum violations of the new inequalities are exemplified within the Clauser--Horne--Shimony--Holt (CHSH) and Klyachko--Can--Binicio\v{g}lu--Shumovsky (KCBS) scenarios. An extension to the case of nnn-cycle scenario is briefly mentioned. The new inequalities with conditional qqq-entropies allow to expand a class of probability distributions, for which the nonlocality or contextuality can be detected within entropic formulation. The qqq-entropic inequalities can also be useful in analyzing cases with detection inefficiencies. Using two models of such a kind, we consider some potential advantages of the qqq-entropic formulation.
Bell's theorem in physics, as well as causal discovery in machine learning, both face the problem of deciding whether observed data is compatible with a presumed causal relationship between the variables (for example, a local hidden variable model). Traditionally, Bell inequalities have been used to describe the restrictions imposed by causal structures on marginal distributions. However, some structures give rise to non-convex constraints on the accessible data, and it has recently been noted that linear inequalities on the observable entropies capture these situations more naturally. In this paper, we show the versatility of the entropic approach by greatly expanding the set of scenarios for which entropic constraints are known. For the first time, we treat Bell scenarios involving multiple parties and multiple observables per party. Going beyond the usual Bell setup, we exhibit inequalities for scenarios with extra conditional independence assumptions, as well as a limited amount of shared randomness between the parties. Many of our results are based on a geometric observation: Bell polytopes for two-outcome measurements can be naturally imbedded into the convex cone of attainable marginal entropies. Thus, any entropic inequality can be translated into one valid for probabilities. In some situations the converse also holds, which provides us with a rich source of candidate entropic inequalities.
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).