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

• URL: http://arxiv.org/abs/2603.28973v1
• Date: Mon, 30 Mar 2026 20:18:16 GMT
• Title: Bell's Inequality, Causal Bounds, and Quantum Bayesian Computation: A Unified Framework
• Abstract summary: Bell inequalities characterize the boundary of the local-realist correlation polytope.<n>Quantum mechanics exceeds this boundary through non-commutativity.<n>We show that the instrumental inequality, the Balke--Pearl linear programming, and the Tian--Pearl probabilities of causation all arise as facets of the same marginal compatibility polytope.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• 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.

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