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

Nick Polson, Vadim Sokolov, Daniel Zantedeschi

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.

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

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 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 . 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.

Paper Structure

This paper contains 41 sections, 19 theorems, 31 equations, 2 figures, 3 tables.

Table of Contents

  1. Introduction
  2. The Classical Feasibility Polytope
  3. Fréchet--Hoeffding Bounds
  4. The Bell--Boole Inequality as a Fréchet Bound
  5. Fine's Theorem: The Pivot
  6. The CHSH Polytope
  7. Causal Inference as Polytope Geometry
  8. Pearl's Instrumental Inequality
  9. Balke--Pearl Bounds via Linear Programming
  10. Tian--Pearl: Probabilities of Causation
  11. Manski's Partial Identification
  12. Kolmogorov Complexity and Entropic Bell Inequalities
  13. The Algorithmic Lens
  14. The Algorithmic Bell Inequality
  15. The Kakeya Connection
  16. ...and 26 more sections

Key Result

Theorem 1

The set of feasible joint distributions $\mathcal{F}(F_0, F_1)$ is the convex set bounded by for all $(u,v) \in [0,1]^2$. The upper bound $M$ is the comonotone coupling; the lower bound $W$ is the countermonotone coupling. $\blacktriangleleft$$\blacktriangleleft$

Figures (2)

  • Figure 1: Cross-section of the correlation space for the CHSH scenario, showing the three nested convex bodies. The local-realist polytope (inner diamond, $|S| \leq 2$) is contained in the quantum set (circle, $|S| \leq 2\sqrt{2}$), which is contained in the no-signaling polytope (outer diamond, $|S| \leq 4$). Here $S = \langle AB\rangle + \langle AB'\rangle + \langle A'B\rangle - \langle A'B'\rangle$ and $S' = \langle AB\rangle - \langle AB'\rangle + \langle A'B\rangle + \langle A'B'\rangle$ are two independent CHSH combinations.
  • Figure 2: The Bell--IV correspondence. Left: Bell scenario---Alice chooses setting $x$, Bob chooses $y$, outcomes $a,b$ depend on settings and shared hidden variable $\lambda$. Right: instrumental variable model---instrument $Z$ affects treatment $X$, which affects outcome $Y$, with latent confounder $U$. The mapping is: measurement setting $\leftrightarrow$ instrument, Alice's outcome $\leftrightarrow$ treatment, Bob's outcome $\leftrightarrow$ observed outcome, hidden variable $\leftrightarrow$ confounder.

Theorems & Definitions (29)

  • Theorem 1: Fréchet, 1951
  • Proposition 2: Probabilistic reformulation
  • Theorem 3: Fine, 1982
  • Definition 4
  • Theorem 5: Pearl, 1995
  • Theorem 6: Structural equivalence; pearl1995testabilitybalke1997bounds
  • Theorem 7: Balke--Pearl, 1997
  • Definition 8
  • Theorem 9: Tian--Pearl, 2000
  • Theorem 10: Manski, 1990
  • ...and 19 more