Formal equivalence between algorithmic/entropic formulations and classical Bell–causal equivalences

Establish whether a formal equivalence (or a rigorous non-equivalence) holds between the algorithmic Kolmogorov-complexity, entropic Bell-inequality, and quantum Bayesian computation formulations of non-classical correlations and the classical probabilistic/causal equivalences (local realism, Fine’s theorem, Fréchet feasibility, and the instrumental inequality).

Background

The paper synthesizes multiple mathematical lenses for non-classical correlations: classical local-realist/causal constraints (Fine’s theorem, Fréchet feasibility, instrumental inequality) and algorithmic/entropic/computational perspectives (Kolmogorov complexity, entropic inequalities, and QBC). Items 1–4 in the synthesis correspond to proven classical equivalences; items 5–7 are structurally parallel but not formally linked.

The authors explicitly identify the absence of a proven bridge between these sets of formulations as an open problem, seeking a rigorous translation or proof of equivalence (or a proof of impossibility).

References

Items 5--7 exhibit the same structural feature—failure of classical factorization—across three distinct domains (algorithms, entropy, computation), but formal equivalence between these layers and items 1--4 remains an open problem.

— Bell's Inequality, Causal Bounds, and Quantum Bayesian Computation: A Unified Framework  (2603.28973 - Polson et al., 30 Mar 2026) in Section 6, Main Synthesis: Layers of Equivalence and Analogy

However, the satisfaction of Eq.(32) does not generally permits a local hidden variable notion as no generalised analogy of Fine's theorem has yet been established.

— No violation on a generalisation of Leggett-Garg inequality and Bell-CHSH inequality with extended probability  (2609.24169 - Kajonsombat et al., 21 Sep 2026) in Section IV, paragraph following the definition of generalized Local realism