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arXiv:1404.4812 · Communications in Mathematical Physics

Beyond Bell’s Theorem II: Scenarios with Arbitrary Causal Structure

T. A. Fritz

Published Nov 5, 2015• 105 citations
arXivHTMLPDF
DOI
10.1007/s00220-015-2495-5

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It is a recent realization that many of the concepts and tools of causal discovery in machine learning are highly relevant to problems in quantum information, in particular quantum nonlocality. The crucial ingredient in the connection between both fields is the mathematical theory of causality, allowing for the representation of arbitrary causal structures and providing a rigorous tool to reason about probabilistic causation. Indeed, Bell's theorem concerns a very particular kind of causal structure and Bell inequalities are a special case of linear constraints following from such models. It is thus natural to look for generalizations involving more complex Bell scenarios. The problem, however, relies on the fact that such generalized scenarios are characterized by polynomial Bell inequalities and no current method is available to derive them beyond very simple cases. In this work, we make a significant step in that direction, providing a new, general, and conceptually clear method for the derivation of polynomial Bell inequalities in a wide class of scenarios. We also show how our construction can be used to allow for relaxations of causal constraints and naturally gives rise to a notion of nonsignaling in generalized Bell networks.
Seen from the modern lens of causal inference, Bell's theorem is nothing other than the proof that a specific classical causal model cannot explain quantum correlations.It is thus natural to move beyond Bell's paradigmatic scenario and consider different causal structures.For the specific case of three observable variables, it is known that there are three nontrivial causal networks.Two of those are known to give rise to quantum nonclassicality: the instrumental and the triangle scenarios.Here we analyze the third and remaining one, which we name the Evans scenario, akin to the causal structure underlying the entanglement-swapping experiment.We prove a number of results about this elusive scenario and introduce new and efficient computational tools for its analysis that can also be adapted to deal with more general causal structures.We do not solve its main open problem-whether quantum nonclassical correlations can arise from it-but give a significant step in this direction by proving that postquantum correlations, analogous to the paradigmatic Popescu-Rohrlich box, do violate the constraints imposed by a classical description of the Evans causal structure.
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.