Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2505.24010

Simplicial methods in the resource theory of contextuality

Aziz Kharoof, Cihan Okay

Published May 29, 2025• 0 citations
arXivHTMLPDF

Abstract

We develop a resource theory of contextuality within the framework of symmetric monoidal categories, extending recent simplicial approaches to quantum contextuality. Building on the theory of simplicial distributions, which integrates homotopy-theoretic structures with probability, we introduce event scenarios as a functorial generalization of presheaf-theoretic measurement scenarios and prove their equivalence to bundle scenarios via the Grothendieck construction. We define symmetric monoidal structures on these categories and extend the distribution functor to a stochastic setting, yielding a resource theory that generalizes the presheaf-theoretic notion of simulations. Our main result characterizes convex maps between simplicial distributions in terms of non-contextual distributions on a corresponding mapping scenario, enhancing and extending prior results in categorical quantum foundations.

DOI
10.48550/arxiv.2505.24010

Related papers

Simplicial distributions, convex categories, and contextuality

Aziz Kharoof, Cihan Okay

The data of a physical experiment can be represented as a presheaf of probability distributions.A striking feature of quantum theory is that those probability distributions obtained in quantum mechanical experiments do not always admit a joint probability distribution, a celebrated observation due to Bell.Such distributions are called contextual.Simplicial distributions are combinatorial models that extend presheaves of probability distributions by elevating sets of measurements and outcomes to spaces.Contextuality can be defined in this generalized setting.This paper introduces the notion of convex categories to study simplicial distributions from a categorical perspective.Simplicial distributions can be given the structure of a convex monoid, a convex category with a single object, when the outcome space has the structure of a group.We describe contextuality as a monoid-theoretic notion by introducing a weak version of invertibility for monoids.Our main result is that a simplicial distribution is noncontextual if and only if it is weakly invertible.Similarly, strong contextuality and contextual fraction can be characterized in terms of invertibility in monoids.Finally, we show that simplicial homotopy can be used to detect extremal simplicial distributions refining the earlier methods based on Čech cohomology and the cohomology of groups.The authors would like to thank the anonymous referee for useful comments that significantly improved the exposition of the paper, and Walker H. Stern and Redi Haderi for enlightening discussions.

arXiv:2211.00571•Theory and applications of categories•Jan 1, 2025•2 citations

A bundle perspective on contextuality: Empirical models and simplicial distributions on bundle scenarios

Rui Soares Barbosa, Aziz Kharoof, Cihan Okay

This paper provides a bundle perspective to contextuality by introducing new categories of contextuality scenarios based on bundles of simplicial complexes and simplicial sets. The former approach generalizes earlier work on the sheaf-theoretic perspective on contextuality, and the latter extends simplicial distributions, a more recent approach to contextuality formulated in the language of simplicial sets. After constructing our bundle categories, we also construct functors that relate them and natural isomorphisms that allow us to compare the notions of contextuality formulated in two languages. We are motivated by applications to the resource theory of contextuality, captured by the morphisms in these categories. In this paper, we develop the main formalism and leave applications to future work.

arXiv:2308.06336•Aug 11, 2023•0 citations

Simplicial quantum contextuality

Cihan Okay, Aziz Kharoof, Selman Ipek

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.

arXiv:2204.06648•Quantum•May 22, 2023•10 citations

Possibilistic collapse and extremality of simplicial distributions

Aziz Kharoof, Cihan Okay

Consistent families of locally defined probability distributions that do not admit a joint global distribution are known as contextual, with primary examples arising in quantum theory. In this paper, we study such families of distributions using the theory of simplicial distributions, and further develop the theory for possibilistic distributions defined over the Boolean semiring. We characterize possibilistic collapses of simplicial distributions geometrically using bundle scenarios. Using this characterization together with a new connectivity condition on the total space of a bundle scenario, we provide a criterion for detecting extremal simplicial distributions. In parallel, we develop an analogous theory for presheaves on simplicial complexes, describe possibilistic collapses of empirical models on them using event scenarios together with a categorical extremality condition, and relate the two frameworks via a comparison result. We provide examples of contextual simplicial distributions that arise from our criteria on scenarios of interest in quantum foundations, such as Bell scenarios and boundaries of standard simplices, the latter connecting to Vorob'ev's classical theorem on acyclic complexes.

math.CTCross-listedarXiv:2607.02754•Jul 6, 2026•0 citations

Twisted simplicial distributions

Cihan Okay, Walker H. Stern

We introduce a theory of twisted simplicial distributions on simplicial principal bundles, which allow us to capture Bell's non-locality, and the more general notion of quantum contextuality. We leverage the classical theory of simplicial principal bundles, as well as structures on categories of such bundles, to provide powerful computational tools for analyzing twisted distributions in terms of both direct constructions in simplicial sets and techniques from homological algebra. We use these techniques to analyze our key examples: quantum distributions and operator-theoretic polytopes used in the classical simulation of quantum computation.

arXiv:2403.19808•Mar 28, 2024•0 citations

The geometry of simplicial distributions on suspension scenarios

Aziz Kharoof

Quantum measurements often exhibit non-classical features, such as contextuality, which generalizes Bell's non-locality and serves as a resource in various quantum computation models. Existing frameworks have rigorously captured these phenomena, and recently, simplicial distributions have been introduced to deepen this understanding. The geometrical structure of simplicial distributions can be seen as a resource for applications in quantum information theory. In this work, we use topological foundations to study this geometrical structure, leveraging the fact that, in this simplicial framework, measurements and outcomes are represented as spaces. This allows us to depict contextuality as a topological phenomenon. We show that applying the cone construction to the measurement space makes the corresponding non-signaling polytope equal to the join of $m$ copies of the original polytope, where $m$ is the number of possible outcomes per measurement. Then we glue two copies of cone measurement spaces to obtain a suspension measurement space. The decomposition done for simplicial distributions on a cone measurement space provides deeper insights into the geometry of simplicial distributions on a suspension measurement space and aids in characterizing the contextuality there. Additionally, we apply these results to derive a new type of Bell inequalities (inequalities that determine the set of local joint probabilities/non-contextual simplicial distributions) and to offer a mathematical explanation for certain contextual vertices from the literature.

arXiv:2412.10963•Dec 14, 2024•0 citations
We develop a resource theory of contextuality within the framework of symmetric monoidal categories, extending recent simplicial approaches to quantum contextuality. Building on the theory of simplicial distributions, which integrates homotopy-theoretic structures with probability, we introduce event scenarios as a functorial generalization of presheaf-theoretic measurement scenarios and prove their equivalence to bundle scenarios via the Grothendieck construction. We define symmetric monoidal structures on these categories and extend the distribution functor to a stochastic setting, yielding a resource theory that generalizes the presheaf-theoretic notion of simulations. Our main result characterizes convex maps between simplicial distributions in terms of non-contextual distributions on a corresponding mapping scenario, enhancing and extending prior results in categorical quantum foundations.
The data of a physical experiment can be represented as a presheaf of probability distributions.A striking feature of quantum theory is that those probability distributions obtained in quantum mechanical experiments do not always admit a joint probability distribution, a celebrated observation due to Bell.Such distributions are called contextual.Simplicial distributions are combinatorial models that extend presheaves of probability distributions by elevating sets of measurements and outcomes to spaces.Contextuality can be defined in this generalized setting.This paper introduces the notion of convex categories to study simplicial distributions from a categorical perspective.Simplicial distributions can be given the structure of a convex monoid, a convex category with a single object, when the outcome space has the structure of a group.We describe contextuality as a monoid-theoretic notion by introducing a weak version of invertibility for monoids.Our main result is that a simplicial distribution is noncontextual if and only if it is weakly invertible.Similarly, strong contextuality and contextual fraction can be characterized in terms of invertibility in monoids.Finally, we show that simplicial homotopy can be used to detect extremal simplicial distributions refining the earlier methods based on Čech cohomology and the cohomology of groups.The authors would like to thank the anonymous referee for useful comments that significantly improved the exposition of the paper, and Walker H. Stern and Redi Haderi for enlightening discussions.
This paper provides a bundle perspective to contextuality by introducing new categories of contextuality scenarios based on bundles of simplicial complexes and simplicial sets. The former approach generalizes earlier work on the sheaf-theoretic perspective on contextuality, and the latter extends simplicial distributions, a more recent approach to contextuality formulated in the language of simplicial sets. After constructing our bundle categories, we also construct functors that relate them and natural isomorphisms that allow us to compare the notions of contextuality formulated in two languages. We are motivated by applications to the resource theory of contextuality, captured by the morphisms in these categories. In this paper, we develop the main formalism and leave applications to future work.
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.
Consistent families of locally defined probability distributions that do not admit a joint global distribution are known as contextual, with primary examples arising in quantum theory. In this paper, we study such families of distributions using the theory of simplicial distributions, and further develop the theory for possibilistic distributions defined over the Boolean semiring. We characterize possibilistic collapses of simplicial distributions geometrically using bundle scenarios. Using this characterization together with a new connectivity condition on the total space of a bundle scenario, we provide a criterion for detecting extremal simplicial distributions. In parallel, we develop an analogous theory for presheaves on simplicial complexes, describe possibilistic collapses of empirical models on them using event scenarios together with a categorical extremality condition, and relate the two frameworks via a comparison result. We provide examples of contextual simplicial distributions that arise from our criteria on scenarios of interest in quantum foundations, such as Bell scenarios and boundaries of standard simplices, the latter connecting to Vorob'ev's classical theorem on acyclic complexes.
We introduce a theory of twisted simplicial distributions on simplicial principal bundles, which allow us to capture Bell's non-locality, and the more general notion of quantum contextuality. We leverage the classical theory of simplicial principal bundles, as well as structures on categories of such bundles, to provide powerful computational tools for analyzing twisted distributions in terms of both direct constructions in simplicial sets and techniques from homological algebra. We use these techniques to analyze our key examples: quantum distributions and operator-theoretic polytopes used in the classical simulation of quantum computation.
Quantum measurements often exhibit non-classical features, such as contextuality, which generalizes Bell's non-locality and serves as a resource in various quantum computation models. Existing frameworks have rigorously captured these phenomena, and recently, simplicial distributions have been introduced to deepen this understanding. The geometrical structure of simplicial distributions can be seen as a resource for applications in quantum information theory. In this work, we use topological foundations to study this geometrical structure, leveraging the fact that, in this simplicial framework, measurements and outcomes are represented as spaces. This allows us to depict contextuality as a topological phenomenon. We show that applying the cone construction to the measurement space makes the corresponding non-signaling polytope equal to the join of mmm copies of the original polytope, where mmm is the number of possible outcomes per measurement. Then we glue two copies of cone measurement spaces to obtain a suspension measurement space. The decomposition done for simplicial distributions on a cone measurement space provides deeper insights into the geometry of simplicial distributions on a suspension measurement space and aids in characterizing the contextuality there. Additionally, we apply these results to derive a new type of Bell inequalities (inequalities that determine the set of local joint probabilities/non-contextual simplicial distributions) and to offer a mathematical explanation for certain contextual vertices from the literature.