Entropic Inequalities and Marginal Problems

Abstract

A marginal problem asks whether a given family of marginal distributions for some set of random variables arises from some joint distribution of these variables. Here we point out that the existence of such a joint distribution imposes non-trivial conditions already on the level of Shannon entropies of the given marginals. These entropic inequalities are necessary (but not sufficient) criteria for the existence of a joint distribution. For every marginal problem, a list of such Shannon-type entropic inequalities can be calculated by Fourier-Motzkin elimination, and we offer a software interface to a Fourier-Motzkin solver for doing so. For the case that the hypergraph of given marginals is a cycle graph, we provide a complete analytic solution to the problem of classifying all relevant entropic inequalities, and use this result to bound the decay of correlations in stochastic processes. Furthermore, we show that Shannon-type inequalities for differential entropies are not relevant for continuous-variable marginal problems; non-Shannon-type inequalities are, both in the discrete and in the continuous case. In contrast to other approaches, our general framework easily adapts to situations where one has additional (conditional) independence requirements on the joint distribution, as in the case of graphical models. We end with a list of open problems. A complementary article discusses applications to quantum nonlocality and contextuality.

26 pages, 3 figures

References in corpus (9)

• Device-independent quantum key distribution secure against collective attacks
• A simple test for hidden variables in spin-1 system
• Beyond Bell's Theorem: Correlation Scenarios
• All noncontextuality inequalities for the n-cycle scenario
• An entropic approach to local realism and noncontextuality
• (Non-)Contextuality of Physical Theories as an Axiom
• Coherent states, entanglement, and geometric invariant theory
• E. Specker: "The logic of non-simultaneously decidable propositions" (1960)
• Information-theoretic inference of common ancestors

Cited by in corpus (55)

• Bell nonlocality in networks
• Kochen-Specker Contextuality
• A Combinatorial Approach to Nonlocality and Contextuality
• All noncontextuality inequalities for the n-cycle scenario
• Information-Theoretic Implications of Quantum Causal Structures
• Theory-independent limits on correlations from generalised Bayesian networks
• Polynomial Bell inequalities
• A unifying framework for relaxations of the causal assumptions in Bell's theorem
• The Inflation Technique for Causal Inference with Latent Variables
• Causal Compatibility Inequalities Admitting Quantum Violations in the Triangle Structure
• Causal structures from entropic information: Geometry and novel scenarios
• All the noncontextuality inequalities for arbitrary prepare-and-measure experiments with respect to any fixed sets of operational equivalences
• Device-Independent Tests of Entropy
• Entropic inequalities as a necessary and sufficient condition to noncontextuality and locality
• Causal inference via algebraic geometry: feasibility tests for functional causal structures with two binary observed variables
• Macrorealism from entropic Leggett-Garg inequalities
• Entropic nonsignaling correlations
• Compatible quantum correlations: on extension problems for Werner and isotropic states
• Quantum marginal problem and incompatibility
• Analysing causal structures with entropy
• Inferring latent structures via information inequalities
• Experimental nonclassicality in a causal network without assuming freedom of choice
• Non-Shannon inequalities in the entropy vector approach to causal structures
• On an Extension Problem for Density Matrices
• Beyond the Cabello-Severini-Winter framework: Making sense of contextuality without sharpness of measurements
• The entropic approach to causal correlations
• On generalized entropies and information-theoretic Bell inequalities under decoherence
• Semidefinite tests for latent causal structures
• Indistinguishability of causal relations from limited marginals
• Quantum space-time marginal problem: global causal structure from local causal information
• Analysing causal structures in generalised probabilistic theories
• Entropic tests of multipartite nonlocality and state-independent contextuality
• Experimental entropic test of state-independent contextuality via single photons
• Inability of the entropy vector method to certify nonclassicality in linelike causal structures
• On the tightness of correlation inequalities with no quantum violation
• Computational tools for solving a marginal problem with applications in Bell non-locality and causal modeling
• Analysing causal structures using Tsallis entropies
• Monogamy of Temporal Correlations: Witnessing non-Markovianity Beyond Data Processing
• Entropic No-Disturbance as a Physical Principle
• Quantum Causal Structure and Quantum Thermodynamics
• Quantum realism and quantum surrealism
• A Combinatorial Solution to Causal Compatibility
• On state vs. channel quantum extension problems: exact results for UxUxU symmetry
• Tests for quantum contextuality in terms of $q$-entropies
• On the complexity of finding the maximum entropy compatible quantum state
• On the insufficiency of entropic inequalities for detecting non-classicality in the Bell causal structure
• The convex class of realisable unit covariances
• Constraints on Multipartite Quantum Entropies
• The quantum marginal independence problem
• Experimental Device-Independent Tests of Classical and Quantum Entropy
• Causal Networks and Freedom of Choice in Bell's Theorem
• Anomalous non-classicality via correlative and entropic Bell inequalities
• Random Measurable Sets and Covariogram Realisability Problems
• Quantum contextuality from measurement invasiveness
• An exactly solvable ansatz for statistical mechanics models