Computational tools for solving a marginal problem with applications in Bell non-locality and causal modeling

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.

References in corpus (13)

• Beyond Bell's Theorem: Correlation Scenarios
• Bell Correlations in a Bose-Einstein Condensate
• Information-Theoretic Implications of Quantum Causal Structures
• Theory-independent limits on correlations from generalised Bayesian networks
• Nonlocal correlations in the star-network configuration
• A unifying framework for relaxations of the causal assumptions in Bell's theorem
• Maximal violation of n-locality inequalities in a star-shaped quantum network
• Experimental non-locality in a quantum network
• Analysing causal structures with entropy
• Which causal structures might support a quantum-classical gap?
• The entropic approach to causal correlations
• Non-Shannon inequalities in the entropy vector approach to causal structures
• Indistinguishability of causal relations from limited marginals

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• The Inflation Technique for Causal Inference with Latent Variables
• Analysing causal structures using Tsallis entropies
• Classifying Causal Structures: Ascertaining when Classical Correlations are Constrained by Inequalities
• Strength of the nonlocality of two-qubit entangled state and its applications