Bidirectional insights between classical and quantum causal inference

Abstract

Causal inference is a sub-area of statistics that investigates the causal explanations underlying observed data. The last decade has seen growing recognition of its applications to quantum physics, which include the use of causal inference to certify quantumness through violations of Bell-like inequalities. This connection places quantum physicists in a unique position to contribute to classical causal inference by addressing open problems whose solutions benefit both fields. This thesis highlights this bidirectional interplay of fields, presenting both the study of foundational problems in classical causal inference from the perspective of quantum physics and the application of these results to quantum research.

An important problem in causal inference is attesting when two causal structures are in principle indistinguishable from data obtained under a given probing scheme on the visible variables. For example, passive observations on two variables cannot distinguish between a direct causal relation and a shared latent common cause. However, they {\em can} be distinguished by interventional probing schemes. The first question we address here is which causal structures remain indistinguishable even under the most informative interventional probing schemes. We derive a necessary and sufficient condition for such indistinguishability. We then consider the same question under access to passive observations only, compiling all known sufficient rules for indistinguishability and applying them to causal structures with three and four visible nodes, though a complete characterization remains open. Then, we extend these results to scenarios involving selection bias.

The works described above also help identify which causal structures impose nontrivial inequality constraints on their classically realizable distributions; an example is the Bell causal structure, that imposes Bell inequalities. This has particular interest for quantum physicists, because inequality constraints are a prerequisite for a causal structure to have a quantum-classical gap (QC gap), that is, for it to explain more distributions when its latent nodes are associated with quantum systems than when its latent nodes are associated with classical variables. This is important because, when we know that a phenomenon is governed by a causal structure that presents a QC gap, observing a probability distribution over the observed variables that cannot be explained using classical latent variables in that structure certifies that the phenomenon involves genuine nonclassicality. Thus, after identifying which causal structures could potentially exhibit a QC gap, we investigate which actually do. Using techniques developed here, we show that hundreds of new causal structures have a QC gap, while previously only around a dozen examples were known.