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

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.

arXiv comment14+7 pages, 3 figures, v2: new results added and parts of the text restructured, v3: version accepted for publication (title differs from published version due to editorial convention)

References in corpus (19)

• Random Numbers Certified by Bell's Theorem
• Bell inequalities for arbitrarily high dimensional systems
• No Signalling and Quantum Key Distribution
• A Relevant Two Qubit Bell Inequality Inequivalent to the CHSH Inequality
• Private Randomness Expansion With Untrusted Devices
• Information theoretical properties of Tsallis entropies
• An entropic approach to local realism and noncontextuality
• Entropic Inequalities and Marginal Problems
• Quantifying Bell: the Resource Theory of Nonclassicality of Common-Cause Boxes
• Looking for symmetric Bell inequalities
• Entropic inequalities as a necessary and sufficient condition to noncontextuality and locality
• Analysing causal structures with entropy
• Bell Inequalities From No-Signalling Distributions
• Non-Shannon inequalities in the entropy vector approach to causal structures
• Information-theoretic Bell inequalities based on Tsallis entropy
• Deriving Tight Bell Inequalities for 2 Parties with Many 2-valued Observables from Facets of Cut Polytopes
• Inability of the entropy vector method to certify nonclassicality in linelike causal structures
• Strengthening the Bell Theorem: conditions to falsify local realism in an experiment
• Analysing causal structures using Tsallis entropies

Cited by in corpus (2)

• Analysing causal structures in generalised probabilistic theories
• Distinguishing quantum causal scenarios with indistinguishable classical analogs: The significance of intermediate latents