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

2026-03-30

AI Summary

This work uncovers deep structural commonalities among Bell inequalities, instrumental variable bounds in causal inference, and quantum Bayesian computation. By constructing a unified framework based on marginal compatibility polytopes, it establishes a structural equivalence between Bell’s local hidden-variable models and instrumental variable models, extending this correspondence to algorithmic, entropic, and semidefinite programming formulations. The study introduces, for the first time, an exact dictionary linking quantum information, causal econometrics, and Bayesian computation, demonstrating—through Fine’s theorem, CHSH inequalities, Balke–Pearl linear programming, and the NPA hierarchy—that both Bell violations and causal bounds originate from a shared geometric structure. Leveraging the Born/Bayes rule duality, the framework enables polynomial acceleration of Bayesian posterior inference and offers a novel architecture for quantum function approximation.

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 $λ$. 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.

Problem

• Bell's Inequality
• Causal Inference
• Local Hidden-Variable Model
• Marginal Compatibility Polytope
• Quantum Bayesian Computation

Innovation

• Bell inequalities
• causal inference
• quantum Bayesian computation
• non-commutativity
• NPA hierarchy