Classifying Causal Structures: Ascertaining when Classical Correlations are Constrained by Inequalities

TL;DR

This paper develops methods to identify causal structures that impose inequality constraints on observed variables, crucial for understanding quantum correlations and their classical limitations.

Contribution

It introduces techniques using d-separation, e-separation, and supports to classify causal scenarios and confirms the exhaustiveness of the HLP condition for up to four variables.

Findings

Most scenarios with only equality constraints are detected by HLP condition.

All but three of the thousands of scenarios with up to four variables are resolved.

HLP condition is likely exhaustive for classifying such causal structures.

Abstract

The classical causal relations between a set of variables, some observed and some latent, can induce both equality constraints (typically conditional independences) as well as inequality constraints (Instrumental and Bell inequalities being prototypical examples) on their compatible distribution over the observed variables. Enumerating a causal structure's implied inequality constraints is generally far more difficult than enumerating its equalities. Furthermore, only inequality constraints ever admit violation by quantum correlations. For both those reasons, it is important to classify causal scenarios into those which impose inequality constraints versus those which do not. Here we develop methods for detecting such scenarios by appealing to d-separation, e-separation, and incompatible supports. Many (perhaps all?) scenarios with exclusively equality constraints can be detected via a condition articulated by Henson, Lal and Pusey (HLP). Considering all scenarios with up to 4 observed variables, which number in the thousands, we are able to resolve all but three causal scenarios, providing evidence that the HLP condition is, in fact, exhaustive.