Abstract Cycle scenarios are a significant class of contextuality scenarios, with the Clauser–Horne–Shimony–Holt scenario being a notable example. While binary outcome measurements in these scenarios are well understood, the generalization to arbitrary outcomes remains less explored, except in specific cases. In this work, we employ homotopical methods in the framework of simplicial distributions to characterize all contextual vertices of the non-signaling polytope corresponding to cycle scenarios with arbitrary outcomes. Additionally, our techniques utilize the bundle perspective on contextuality and the decomposition of measurement spaces. This enables us to extend beyond scenarios formed by gluing cycle scenarios and describe contextual extremal simplicial distributions in these generalized contexts.
Abstract Cycle scenarios are a significant class of contextuality scenarios, with the Clauser–Horne–Shimony–Holt scenario being a notable example. While binary outcome measurements in these scenarios are well understood, the generalization to arbitrary outcomes remains less explored, except in specific cases. In this work, we employ homotopical methods in the framework of simplicial distributions to characterize all contextual vertices of the non-signaling polytope corresponding to cycle scenarios with arbitrary outcomes. Additionally, our techniques utilize the bundle perspective on contextuality and the decomposition of measurement spaces. This enables us to extend beyond scenarios formed by gluing cycle scenarios and describe contextual extremal simplicial distributions in these generalized contexts.
Simplicial distributions are combinatorial models describing distributions on spaces of measurements and outcomes that generalize nonsignaling distributions on contextuality scenarios. This paper studies simplicial distributions on two-dimensional measurement spaces by introducing new topological methods. Two key ingredients are a geometric interpretation of Fourier–Motzkin elimination and a technique based on the collapsing of measurement spaces. Using the first one, we provide a new proof of Fine’s theorem characterizing noncontextual distributions in N-cycle scenarios. Our approach goes beyond these scenarios and can describe noncontextual distributions in scenarios obtained by gluing cycle scenarios of various sizes. The second technique is used for detecting contextual vertices and deriving new Bell inequalities. Combined with these methods, we explore a monoid structure on simplicial distributions.
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.
Contextuality can be understood as the impossibility to construct a globally consistent description of a model even if there is local agreement. In particular, quantum models present this property. We can describe contextuality with the bundle approach, where the scenario is represented as a simplicial complex, the fibers being the sets of outcomes, and contextuality as the non-existence of global section in the measure bundle. Using the generalization to non-finite outcome fibers, we built in details the concept of measure bundle, demonstrating the Fine-Abramsky-Brandenburger theorem for the bundle formalism. We introduce a hierarchy called n-contextuality to explore the dependence of contextual behavior of a model to the topology of the scenario, following the construction of it as a simplicial complex. With it we exemplify the dependence on higher homology groups and show that GHZ models, thus quantum theory, has all levels of the hierarchy. Also, we give an example of how non-trivial topology of the scenario result an increase of contextual behavior. For the first level of the hierarchy, we construct the concept of connection through Markov operators for the measure bundle. Taking the case of equal fibers we can identify the outcomes as the basis of a vector space, that transform according to a group extracted from the connection. We thus show that contextuality has a relationship with the non-triviality of the holonomy group in the respective frame bundle.
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.
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.
Contextuality can be understood as the impossibility to construct a globally consistent description of a model even if there is local agreement. In particular, quantum models present this property. We can describe contextuality with the fibration approach, where the scenario is represented as a simplicial complex, the fibers being the sets of outcomes, and contextuality as the non-existence of a global section in the measure fibration, allowing direct representation and formalization of the already used bundle diagrams. Using the generalization to continuous outcome fibers, we built the concept of measure fibration, showing the Fine-Abramsky-Brandenburger theorem for the fibration formalism in the case of non-finite fibers. By the Voroby'ev theorem, we argue that the dependence of contextual behavior of a model to the topology of the scenario is an open problem. We introduce a hierarchy of contextual behavior to explore it, following the construction of the simplicial complex. GHZ models show that quantum theory has all levels of the hierarchy, and we exemplify the dependence on higher homotopical groups by the tetraedron scenario, where non-trivial topology implies an increase of contextual behavior for this case. For the first level of the hierarchy, we construct the concept of connection through Markov operators for the measure bundle using the measure on fibers of contexts with two measurements and taking the case of equal fibers we can identify the outcome space as the basis of a vector space, that transform according to a group extracted from the connection. With this, it is possible to show that contextuality at the level of contexts with two measurements has a relationship with the non-triviality of the holonomy group in the frame bundle. We give examples and treat disturbing models through transition functions, generalizing the holonomy.