Analysis of the entropy vector approach to distinguish classical and quantum causal structures

Causal structures play an important role in the foundations of quantum mechanics because of Bell's theorem, which states that particular observed correlations could not have been generated by classical means within a particular causal structure. While in the simplest causal structures, the problem of certifying non-classicality is well-understood, in more complicated cases it is not. We consider the question of whether the joint entropies of a set of observed random variables can lead to useful certificates of non-classicality. We find that for a family of causal structures that include the usual bipartite Bell structure they do not, in spite of the existence of non-classical correlations. We furthermore find that for many causal structures non-Shannon entropic inequalities give additional constraints on the sets of possible entropy vectors in the classical case and hence lead to tighter approximations of the set of realisable entropy vectors.