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arXiv:2604.26916 · Journal of Theoretical Experimental and Applied Physics

Contextuality, Locality, and the Interpretation of Bell Violations

Partha Ghose

quant-phSubmitted Apr 29, 2026• Announced Apr 30, 2026• 0 citations• Revised Jul 1, 2026 (v3)
v1 · Apr 29, 2026v2 · May 2, 2026v3 · Jul 1, 2026
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Abstract

The ontological significance of violations of Bell inequalities is reexamined. Bell's theorem admits several mathematically equivalent formulations that support different interpretive perspectives. The present paper develops one such perspective using contextuality, the sheaf-theoretic framework of Abramsky and Brandenburger, and Nelson's stochastic mechanics.

Comments
10 pages, no figures; revised to strengthen the claim; new references added

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The ontological significance of violations of Bell inequalities is reexamined. Bell's theorem admits several mathematically equivalent formulations that support different interpretive perspectives. The present paper develops one such perspective using contextuality, the sheaf-theoretic framework of Abramsky and Brandenburger, and Nelson's stochastic mechanics.
1. Einstein A. Can quantum-mechanical description of physical reality be considered complete? / A. Einstein, B. Podolsky, N. Rosen // Phys. Rev. 47 — 1935 — 777. 2. Bell J.S. On the Einstein Podolsky Rosen paradox / Bell J.S. // Physics (Long Island City, N.Y.) 1 — 1964 — 195. 3. Brunner N. Bell nonlocality / N. Brunner, D. Cavalcanti, S. Pironio, V. Scarani, S. Wehner // Rev. Mod. Phys. 86 — 2014 — 419. 4. Fine. A Hidden variables, joint probability, and the Bell inequalities / A. Fine // Phys. Rev. Lett 48 — 1982 — 291. 5. Abramsky S. The sheaf-theoretic structure of non-locality and contextuality / S. Abramsky, A. Brandenburger // New J. Phys. 13 — 2011 — 113036. 6. Farkas G. Theorie der Einfachen Ungleichungen / G. Farkas // Journal für die Reine und Angewandte Mathematik 124 — 1902 — с. 1-27.
The sheaf theoretic description of non-locality and contextuality by Abramsky and Brandenburger sets the ground for a topological study of these peculiar features of quantum mechanics. This viewpoint has been recently developed thanks to sheaf cohomology, which provides a sufficient condition for contextuality of empirical models in quantum mechanics and beyond. Subsequently, a number of studies proposed methods to detect contextuality based on different cohomology theories. However, none of these cohomological descriptions succeeds in giving a full invariant for contextuality applicable to concrete examples. In the present work, we introduce a cohomology invariant for possibilistic and strong contextuality which is applicable to the vast majority of empirical models.
Among the various "no-go theorems" in the foundations of quantum mechanics, Bell inequalities are among the few, which do not implicitly rely on the mathematical structure of quantum mechanics. Instead, Bell inequalities are commonly understood to rest on Kolmogorov's theory of probability and a "locality" assumption. Physical evidence supporting their violation is hence taken as evidence against at least one of these two assumptions. This work provides a comprehensive analysis of the physical debate on Bell inequalities from the point of view of Kolmogorov's theory of probability, focusing on the Bell-Clauser-Horn-Shimony-Holt inequality (BCHSH inequality). The central contribution is the motivation and mathematical study of the notion of contextuality in Bell experiments: in so-called contextual Bell models the detectors themselves behave randomly and the distribution of the random detector parameters depends on the chosen detector settings. It is shown how even "classical" Bell experiments motivate such a description. The need to account for contextuality was pointed out before by de la Peña, Cetto, Brody, Lochak, Bohm, Hiley and others. A broad class of contextual models is proven to satisfy the BCHSH inequality, thereby generalizing a previous result by Gill and Lambare. Nonetheless, even under strong locality requirements, contextual Bell models may violate the BCHSH inequality: based on work by Popescu and Rohrlich, we consider a "proof of concept" example exhibiting the maximum possible violation. Contrary to the much debated "loopholes" in empirical tests of Bell inequalities, contextuality thus reveals a conceptual misunderstanding in the physical conclusions commonly drawn from Bell inequality violations.
Extendability of an empirical model was shown by Abramsky & Brandenburger to correspond in a unified manner to both locality and non-contextuality. We develop their approach by presenting a refinement of the notion of extendability that can also be useful in characterising the properties of sub-models. The refinement is found to have another useful application: it is shown that a particular canonical extension, when well-defined, may be used for the construction of Bell-type models from models of more general kinds in such a way that the constructed model is equivalent to the original in terms of non-locality/contextuality. This is important since on practical and foundational levels, the notion of locality in Bell-type models can more easily be motivated than the corresponding general notion of contextuality. We consider examples of Bell-type models generated from some standard examples of contextual models, including an entire class of Kochen-Specker-like models. This exposes an intriguing relationship between the simplest possible contextual model (the contextual triangle) and Popescu-Rohrlich no-signalling correlations.
Non-locality and contextuality are key features of quantum mechanics that distinguish it from classical physics. We aim to develop a deeper, more structural understanding of these phenomena, underpinned by robust and elegant mathematical theory with a view to providing clarity and new perspectives on conceptual and foundational issues. A general framework for logical non-locality is introduced and used to prove that 'Hardy's paradox' is complete for logical non-locality in all (2,2,l) and (2,k,2) Bell scenarios, a consequence of which is that Bell states are the only entangled two-qubit states that are not logically non-local, and that Hardy non-locality can be witnessed with certainty in a tripartite quantum system. A number of developments of the unified sheaf-theoretic approach to non-locality and contextuality are considered, including the first application of cohomology as a tool for studying the phenomena: we find cohomological witnesses corresponding to many of the classic no-go results, and completely characterise contextuality for large families of Kochen-Specker-like models. A connection with the problem of the existence of perfect matchings in k-uniform hypergraphs is explored, leading to new results on the complexity of deciding contextuality. A refinement of the sheaf-theoretic approach is found that captures partial approximations to locality/non-contextuality and can allow Bell models to be constructed from models of more general kinds which are equivalent in terms of non-locality/contextuality. Progress is made on bringing recent results on the nature of the wavefunction within the scope of the logical and sheaf-theoretic methods. Computational tools are developed for quantifying contextuality and finding generalised Bell inequalities for any measurement scenario which complement the research programme. This also leads to a proof that local ontological models with `negative probabilities' generate the no-signalling polytopes for all Bell scenarios.