2013/10/06 by Rafael Chaves, Lukas Luft, David Gross +1 · 3 citations
Computer Science · Mathematics · Physics and Astronomy · #Artificial intelligence #Bell's theorem #Causal structure #Computer science #Conditional independence #Converse #Discrete mathematics #Geometry #Hidden variable theory #Inequality #Mathematical analysis #Mathematical economics #Mathematics #Observable #Physics #Polytope #Quantum #Quantum Information and Cryptography #Quantum Mechanics and Applications #Quantum entanglement #Quantum mechanics #Randomness #Set (abstract data type) #Statistical Mechanics and Entropy #Statistical physics #Statistics #Theoretical physics #quant-ph
paper · pdf · doi:10.1088/1367-2630/16/4/043001
published as New J. Phys. 16, 043001 (2014) · 20 pages, 7 figures, many cones. Minor revisions, new figures and inequalities
arxiv created 2013/10/06 · openalex publication_date 2014/04/03 · arxiv updated 2014/06/02 · openalex created_date 2020/11/23 · openalex updated_date 2026/08/05
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.