Estimating the volumes of correlations sets in causal networks

TL;DR

This paper investigates the volume of classical versus non-classical correlations in causal networks, revealing limitations of current detection methods and highlighting the potential of interventions to better identify non-classical behaviors.

Contribution

It introduces a volume-based approach to analyze correlations in causal networks and demonstrates the limitations of the inflation technique while showing how interventions improve detection.

Findings

Inflation technique misses significant non-classical correlations.

Interventions substantially enhance non-classicality detection.

Non-convex correlation sets pose challenges for analysis.

Abstract

Causal networks beyond that in the paradigmatic Bell's theorem can lead to new kinds and applications of non-classical behavior. Their study, however, has been hindered by the fact that they define a non-convex set of correlations and only very incomplete or approximated descriptions have been obtained so far, even for the simplest scenarios. Here, we take a different stance on the problem and consider the relative volume of classical or non-classical correlations a given network gives rise to. Among many other results, we show instances where the inflation technique, arguably the most disseminated tool in the community, is unable to detect a significant portion of the non-classical behaviors. Interestingly, we also show that the use of interventions, a central tool in causal inference, can enhance substantially our ability to witness non-classicality.