Simplicial Quantum Contextuality

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.

Featured image: (Left) The CHSH scenario can be organized into a surface topologically equivalent to a punctured torus. Edges with the same label on the boundary are identified. (Right) The PR box on the punctured torus cannot be extended to a distribution on the torus. Nonextendibility is a topological characterization of contextuality.

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Popular summary

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.

► BibTeX data

@article{Okay2023simplicialquantum, doi = {10.22331/q-2023-05-22-1009}, url = {https://doi.org/10.22331/q-2023-05-22-1009}, title = {Simplicial quantum contextuality}, author = {Okay, Cihan and Kharoof, Aziz and Ipek, Selman}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{“{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {7}, pages = {1009}, month = may, year = {2023}<br /> }

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Cited by

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

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

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

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

The above citations are from SAO/NASA ADS (last updated successfully 2023-05-22 12:36:33). The list may be incomplete as not all publishers provide suitable and complete citation data.

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