On the insufficiency of entropic inequalities for detecting non-classicality in the Bell causal structure

TL;DR

This paper demonstrates that entropic inequalities, including those based on Shannon and Tsallis entropies, are insufficient for detecting non-classical correlations in generalized Bell scenarios, highlighting limitations in current non-locality detection methods.

Contribution

It shows that entropic inequalities cannot detect certain non-classical correlations in Bell scenarios with more outcomes, even with natural extensions and post-processing.

Findings

Entropic inequalities fail to detect non-classicality in some generalized Bell scenarios.

Tsallis entropy does not improve detection over Shannon entropy.

The vertex description of non-signalling distributions satisfying CHSH inequalities is provided.

Abstract

Classical and quantum physics impose different constraints on the joint probability distributions of observed variables in a causal structure. These differences mean that certain correlations can be certified as non-classical, which has both foundational and practical importance. Rather than working with the probability distribution itself, it can instead be convenient to work with the entropies of the observed variables. In the Bell causal structure with two inputs and outputs per party, a technique that uses entropic inequalities is known that can always identify non-classical correlations. Here we consider the analogue of this technique in the generalization of this scenario to more outcomes. We identify a family of non-classical correlations in the Bell scenario with two inputs and three outputs per party whose non-classicality cannot be detected through the direct analogue of the previous technique. We also show that use of Tsallis entropy instead of Shannon entropy does not help in this case. Furthermore, we give evidence that natural extensions of the technique also do not help. More precisely, our evidence suggests that even if we allow the observed correlations to be post-processed according to a natural class of non-classicality non-generating operations, entropic inequalities for either the Shannon or Tsallis entropies cannot detect the non-classicality, and hence that entropic inequalities are generally not sufficient to detect non-classicality in the Bell causal structure. In addition, for the bipartite Bell scenario with two inputs and three outputs we find the vertex description of the polytope of non-signalling distributions that satisfy all of the CHSH-type inequalities, which is one of the main regions of investigation in this work.