A corollary to Bell’s theorem

Nick Herbert, Jack Karush

Márton Gömöri, Carl Hoefer

Abstract A widespread view among physicists is that Bell’s theorem rests on an implicit assumption of “classicality,” in addition to locality. According to this understanding, the violation of Bell’s inequalities poses no challenge to locality, but simply reinforces the fact that quantum mechanics is not classical. The paper provides a critical analysis of this view. First we characterize the notion of classicality in probabilistic terms. We argue that classicality thus construed has nothing to do with the validity of classical physics, nor of classical probability theory, contrary to what many believe. At the same time, we show that the probabilistic notion of classicality is not an additional premise of Bell’s theorem, but a mathematical corollary of locality in conjunction with the standard auxiliary assumptions of Bell. Accordingly, any theory that claims to get around the derivation of Bell’s inequalities by giving up classicality, in fact has to give up one of those standard assumptions. As an illustration of this, we look at two recent interpretations of quantum mechanics, Reinhard Werner’s operational quantum mechanics and Robert Griffiths’ consistent histories approach, that are claimed to be local and non-classical, and identify which of the standard assumptions of Bell’s theorem each of them is forced to give up. We claim that while in operational quantum mechanics the Common Cause Principle is violated, the consistent histories approach is conspiratorial. Finally, we relate these two options to the idea of realism, a notion that is also often identified as an implicit assumption of Bell’s theorem.

А. В. Белинский

Krzysztof Rębilas

A fundamental Bell-type inequality is derived by referring solely to laws of logic, tertium non datur and the transitivity axiom, that are used to express the local realism assumption. The general form of the achieved inequality makes it a root of other more specific Bell-type theorems involving additional mathematical formalism and experimental conditions. Its universality is demonstrated explicitly by showing that the most famous theorems provided by Bell, Clauser, Horne, Shimony and Holt (CHSH), Clauser and Horne (CH), Eberhard, d'Espagnat and Maccone are direct corollaries of the general inequality. The method to retrieve Bell's inequalities of different kinds, as being based on a single scheme of reasoning, unifies diversity of proofs presented in the literature. Additionally, some of the known inequalities can be easily achieved in a more general form compared to their original versions and still suitable, or possibly even more convenient, for experimental investigations. By extracting local realism in the form of an explicit logical propositions and making it a cornerstone of a systematic reasoning leading to the universal Bell's inequality, the paper offers a synthetic perspective on Bell-type theorems and their logical basis.

Anton Lorenz Vrba, Richard David Gill

We examine the logical and mathematical structure of Sanctuary's recent paper (\emph{Quantum Rep.}\ 2026, 8, 96), which argues that spin observables in the Bivector Standard Model (BiSM) are not jointly definable on a single Kolmogorov probability space, so that the CHSH quadruple cannot be formed and Bell's and Fine's theorems do not apply. The paper contains two constructions: local response functions of the analyser setting and a shared source phase, and a separate prescription that produces the correlation $-\cos(a-b)$. We show that the first construction, as specified, meets the hypotheses of Bell's theorem. Its outcome maps are defined on a common source domain with a setting-independent distribution, so the non-existence claim (Lemma~3, Corollary~3) does not hold for it, its product correlations obey the CHSH bound, and the paper's own simulation of it yields the Bell triangle. The cosine is obtained instead from a scalar computed jointly from both stations' phases and then digitised; it is not shown to be the correlation of the local outcomes. Execution of the supplementary programs further shows that the four BiSM\_v2 populations do not form the partition stated in the paper, and that the doubled-angle harmonic attributed to the raw coherence data appears only after a sign convention is applied on the second half-cycle.

Kaj Börge Hansen