Theory-independent limits on correlations from generalised Bayesian networks

TL;DR

This paper generalizes Bayesian networks to include non-classical resources like quantum and super-quantum systems, establishing theory-independent constraints and classifying causal structures for different probabilistic theories.

Contribution

It introduces a framework of generalised Bayesian networks that extends classical causal models to arbitrary probabilistic theories, including quantum and beyond.

Findings

Classical conditional independences hold in the generalized framework.

Probabilistic constraints can go beyond classical independences.

Certain causal structures may distinguish classical, quantum, and super-quantum correlations.

Abstract

Bayesian networks provide a powerful tool for reasoning about probabilistic causation, used in many areas of science. They are, however, intrinsically classical. In particular, Bayesian networks naturally yield the Bell inequalities. Inspired by this connection, we generalise the formalism of classical Bayesian networks in order to investigate non-classical correlations in arbitrary causal structures. Our framework of `generalised Bayesian networks' replaces latent variables with the resources of any generalised probabilistic theory, most importantly quantum theory, but also, for example, Popescu-Rohrlich boxes. We obtain three main sets of results. Firstly, we prove that all of the observable conditional independences required by the classical theory also hold in our generalisation; to obtain this, we extend the classical $d$-separation theorem to our setting. Secondly, we find that the theory-independent constraints on probabilities can go beyond these conditional independences. For example we find that no probabilistic theory predicts perfect correlation between three parties using only bipartite common causes. Finally, we begin a classification of those causal structures, such as the Bell scenario, that may yield a separation between classical, quantum and general-probabilistic correlations.