Non-Shannon inequalities in the entropy vector approach to causal structures

Abstract

A causal structure is a relationship between observed variables that in general restricts the possible correlations between them. This relationship can be mediated by unobserved systems, modelled by random variables in the classical case or joint quantum systems in the quantum case. One way to differentiate between the correlations realisable by two different causal structures is to use entropy vectors, i.e., vectors whose components correspond to the entropies of each subset of the observed variables. To date, the starting point for deriving entropic constraints within causal structures are the so-called Shannon inequalities (positivity of entropy, conditional entropy and conditional mutual information). In the present work we investigate what happens when non-Shannon entropic inequalities are included as well. We show that in general these lead to tighter outer approximations of the set of realisable entropy vectors and hence enable a sharper distinction of different causal structures. Since non-Shannon inequalities can only be applied amongst classical variables, it might be expected that their use enables an entropic distinction between classical and quantum causal structures. However, this remains an open question. We also introduce techniques for deriving inner approximations to the allowed sets of entropy vectors for a given causal structure. These are useful for proving tightness of outer approximations or for finding interesting regions of entropy space. We illustrate these techniques in several scenarios, including the triangle causal structure.

23 pages + appendix; v2: minor changes to Section IV A; v3: paper has been significantly shortened, an expanded version of the removed review section can be found in arXiv:1709.08988; v4: version to be published, supplementary information available as ancillary files

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• Private Randomness Expansion With Untrusted Devices
• Beyond Bell's Theorem: Correlation Scenarios
• Information-Theoretic Implications of Quantum Causal Structures
• Theory-independent limits on correlations from generalised Bayesian networks
• Analysing causal structures with entropy
• Inferring latent structures via information inequalities
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Cited by in corpus (17)

• Bell nonlocality in networks
• The Inflation Technique for Causal Inference with Latent Variables
• Causal Compatibility Inequalities Admitting Quantum Violations in the Triangle Structure
• Limits on correlations in networks for quantum and no-signaling resources
• A neural network oracle for quantum nonlocality problems in networks
• Network Quantum Steering
• Quantum entanglement in the triangle network
• Partial self-testing and randomness certification in the triangle network
• Towards a minimal example of quantum nonlocality without inputs
• Semidefinite tests for latent causal structures
• Analysing causal structures in generalised probabilistic theories
• Computational tools for solving a marginal problem with applications in Bell non-locality and causal modeling
• Analysing causal structures using Tsallis entropies
• Noise-robust proofs of quantum network nonlocality
• On the insufficiency of entropic inequalities for detecting non-classicality in the Bell causal structure
• Memory attacks in network nonlocality and self-testing
• Exploring the Local Landscape in the Triangle Network