Bell's Inequality, Causal Bounds, and Quantum Bayesian Computation: A Unified Framework

TL;DR

This paper reveals a unified framework connecting Bell inequalities, causal inference, and quantum Bayesian computation, highlighting their structural similarities and implications for quantum and causal modeling.

Contribution

It establishes a detailed correspondence between quantum Bell inequalities and causal inference bounds, extending to algorithmic, entropic, and computational complexity formulations.

Findings

Bell inequalities and causal bounds share the same polytope structure.

Quantum Bayesian computation exploits non-commutativity for speedups.

The framework creates a dictionary linking quantum information, causal inference, and Bayesian methods.

Abstract

Bell inequalities characterize the boundary of the local-realist correlation polytope -- the set of joint probability distributions achievable by classical hidden-variable models. Quantum mechanics exceeds this boundary through non-commutativity, reaching the Tsirelson bound $2\sqrt{2}$ for CHSH. We show that this polytope structure is not specific to quantum foundations: it appears identically in the causal inference literature, where the instrumental inequality, the Balke--Pearl linear programming bounds, and the Tian--Pearl probabilities of causation all arise as facets of the same marginal compatibility polytope. Fine's theorem -- that CHSH inequalities hold if and only if a joint distribution exists -- is precisely the pivot: the instrumental variable model in causal inference is structurally equivalent to the Bell local hidden-variable model, with the instrument playing the role of the measurement setting and the latent confounder playing the role of the hidden variable $\lambda$. We develop this correspondence in detail, extending it to algorithmic (Kolmogorov complexity) and entropic formulations of Bell inequalities, the NPA semidefinite programming hierarchy, and the MIP$^*$=RE undecidability result. We further show that the Born-rule / Bayes-rule duality underlying quantum Bayesian computation exploits the same non-commutativity that enables Bell violation, providing polynomial speedups for posterior inference. The framework yields a concrete dictionary between quantum information theory, causal econometrics, and Bayesian computation, and suggests new directions including NPA-based quantum causal inference algorithms and quantum architectures for function approximation.