Simplicial quantum contextuality

Cihan Okay, Aziz Kharoof, and Selman Ipek

Department of Mathematics, Bilkent University, Ankara, Turkey

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Abstract

We introduce a new framework for contextuality based on simplicial sets, combinatorial models of topological spaces that play a prominent role in modern homotopy theory. Our approach extends measurement scenarios to consist of spaces (rather than sets) of measurements and outcomes, and thereby generalizes nonsignaling distributions to simplicial distributions, which are distributions on spaces modeled by simplicial sets. Using this formalism we present a topologically inspired new proof of Fine's theorem for characterizing noncontextuality in Bell scenarios. Strong contextuality is generalized suitably for simplicial distributions, allowing us to define cohomological witnesses that extend the earlier topological constructions restricted to algebraic relations among quantum observables to the level of probability distributions. Foundational theorems of quantum theory such as the Gleason's theorem and Kochen--Specker theorem can be expressed naturally within this new language.

Contextuality is a fundamental feature of quantum mechanics in which measurement statistics cannot be reproduced by hidden variable models with pre-existing outcome assignments, a phenomenon which is captured by violations of Bell inequalities. We introduce a new topological framework for contextuality in which simplicial sets, central objects in modern homotopy theory, are used to model measurement statistics as distributions on spaces. Quantum statistics satisfy a principle known as nonsignaling, or nondisturbance. In this simplicial framework this compatibility property is encoded topologically as the gluing of simplices, giving rise to simplicial distributions. Topological notions, such as gluing and extension, can be used to systematize the analysis of contextuality, leading to novel characterizations of Bell inequalities. The theory of simplicial distributions unifies the sheaf-theoretic and the group cohomological approaches to contextuality and goes beyond by allowing for new scenarios and novel proof techniques that cannot be realized in earlier frameworks. Such a topological perspective brings new insight into the study of Bell inequalities.

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[2] Rui Soares Barbosa, Aziz Kharoof, and Cihan Okay, "A bundle perspective on contextuality: Empirical models and simplicial distributions on bundle scenarios", arXiv:2308.06336, (2023).

[3] Kin Ian Lo, Mehrnoosh Sadrzadeh, and Shane Mansfield, "Generalised Winograd Schema and its Contextuality", arXiv:2308.16498, (2023).

[4] Aziz Kharoof and Cihan Okay, "Simplicial distributions, convex categories and contextuality", arXiv:2211.00571, (2022).

[5] Cihan Okay, Ho Yiu Chung, and Selman Ipek, "Mermin polytopes in quantum computation and foundations", arXiv:2210.10186, (2022).

[6] Sivert Aasnæss, "Comparing two cohomological obstructions for contextuality, and a generalised construction of quantum advantage with shallow circuits", arXiv:2212.09382, (2022).

[7] Cihan Okay and Igor Sikora, "Equivariant simplicial distributions and quantum contextuality", arXiv:2310.18135, (2023).

[8] Aziz Kharoof and Cihan Okay, "Homotopical characterization of strongly contextual simplicial distributions on cone spaces", arXiv:2311.14111, (2023).

[9] Ho Yiu Chung, Cihan Okay, and Igor Sikora, "Simplicial techniques for operator solutions of linear constraint systems", arXiv:2305.07974, (2023).

[10] Selman Ipek and Cihan Okay, "The degenerate vertices of the $2$-qubit $\Lambda$-polytope and their update rules", arXiv:2312.10734, (2023).

[11] Cihan Okay, "On the rank of two-dimensional simplicial distributions", arXiv:2312.15794, (2023).

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