Simplicial methods in the resource theory of contextuality

TL;DR

This paper develops a categorical and simplicial framework for resource theory of quantum contextuality, integrating homotopy theory and probability to generalize measurement scenarios and characterize non-contextuality.

Contribution

It introduces a functorial generalization of measurement scenarios, extends the distribution functor to stochastic settings, and characterizes convex maps via non-contextual distributions.

Findings

Equivalence of event and bundle scenarios via Grothendieck construction

Extension of the distribution functor to stochastic settings

Characterization of convex maps through non-contextual distributions

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.