Identical particles and the simplicial category

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.

E. Celeghini, VITIELLO G., Mario Rasetti

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.

D. B. Lichtenberg

Aziz Kharoof, Cihan Okay

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.

D. B. Lichtenberg