Journal
NEW JOURNAL OF PHYSICS
Volume 16, Issue -, Pages -Publisher
IOP Publishing Ltd
DOI: 10.1088/1367-2630/16/4/043001
Keywords
non-locality; marginal scenarios; causal structures; entropic inequalities
Categories
Funding
- Excellence Initiative of the German Federal and State Governments [ZUK 43]
- Research Innovation fund from the University of Freiburg
- open access publication fund of the Albert Ludwig University, Freiburg
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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.
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