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

TL;DR

This paper explores how incorporating non-Shannon entropic inequalities can improve the differentiation of causal structures via entropy vectors, potentially distinguishing classical from quantum causal models.

Contribution

It demonstrates that non-Shannon inequalities tighten the outer bounds of entropy vector sets and introduces methods for deriving inner approximations within causal structures.

Findings

Non-Shannon inequalities lead to tighter outer bounds.

Techniques for inner approximation of entropy vectors are developed.

Application to triangle causal structure illustrates the methods.

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.