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.

翻译：贝尔不等式刻画了局域实在关联多面体的边界——即经典隐变量模型所能实现的联合概率分布集合。量子力学通过非对易性突破该边界，使CHSH不等式达到茨雷尔松界$2\sqrt{2}$。我们证明这一多面体结构并非量子基础领域所独有：它在因果推断文献中以完全相同的形式出现，其中工具不等式、巴尔克-珀尔线性规划界限以及田-珀尔因果概率均作为同一边缘相容多面体的面而产生。法因定理——CHSH不等式成立当且仅当存在联合分布——恰好构成枢轴：因果推断中的工具变量模型在结构上等价于贝尔局域隐变量模型，其中工具扮演测量设置的角色，而潜在混杂变量扮演隐变量$λ$的角色。我们详细阐述这种对应关系，并将其推广至贝尔不等式的算法（科尔莫戈罗夫复杂度）和熵表述、NPA半定规划层级以及MIP$^*$=RE不可判定性结果。我们进一步证明，量子贝叶斯计算所基于的玻恩规则/贝叶斯规则对偶性利用了与突破贝尔不等式相同的非对易性，从而为后验推断提供多项式加速。该框架构建了量子信息论、因果计量经济学与贝叶斯计算之间的具体词典，并提出了包括基于NPA的量子因果推断算法和用于函数逼近的量子架构等新方向。