Abstract
Marginal problems naturally arise in a variety of different fields: basically, the question is whether some marginal/partial information is compatible with a joint probability distribution. To this aim, the characterization of marginal sets via quantifier elimination and polyhedral projection algorithms is of primal importance. In this work, before considering specific problems, we review polyhedral projection algorithms with focus on applications in information theory, and, alongside known algorithms, we also present a newly developed geometric algorithm which walks along the face lattice of the polyhedron in the projection space. One important application of this is in the field of quantum non-locality, where marginal problems arise in the computation of Bell inequalities. We apply the discussed algorithms to discover many tight entropic Bell inequalities of the tripartite Bell scenario as well as more complex networks arising in the field of causal inference. Finally, we analyze the usefulness of these inequalities as nonlocality witnesses by searching for violating quantum states.
Highlights
Starting point of this paper is the marginal problem: given joint distributions of certain subsets of random variables X1, . . . , Xn, are they compatible with the existence of any joint distribution for all these variables? In other words, is it possible to find a joint distribution for all these variables, such that this distribution marginalizes to the given ones? Such a problem naturally arises in several different fields
Marginal problems naturally arise in a variety of different fields: basically, the question is whether some marginal/partial information is compatible with a joint probability distribution
The characterization of the set of correlations/probability distributions of a given marginal scenario is of central relevance in a variety of fields
Summary
Starting point of this paper is the marginal problem: given joint distributions of certain subsets of random variables X1, . . . , Xn, are they compatible with the existence of any joint distribution for all these variables? In other words, is it possible to find a joint distribution for all these variables, such that this distribution marginalizes to the given ones? Such a problem naturally arises in several different fields. The situation is even worse for the study of nonlocality in complex quantum networks, where on the top of local realism one imposes additional constraints [10,11,12,13,14,15,16,17,18,19,20] In this case, the derivation of Bell inequalities involves the characterization of complicated non-convex sets for which even more computationally demanding tools from algebraic geometry [15, 17, 21, 22] seem to be the only viable alternative. For the convenience of the reader, all inequalities found in this work are listed in Appendix XV, and XVI and are available online [57]
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