Extremal simplicial distributions on cycle scenarios with arbitrary outcomes

TL;DR

This paper characterizes all extremal non-signaling distributions in cycle scenarios with arbitrary outcomes using homotopical and bundle methods, extending understanding beyond binary cases and simple gluings.

Contribution

It introduces a homotopical approach to classify extremal simplicial distributions in generalized cycle scenarios with arbitrary outcomes, advancing the theoretical framework of contextuality.

Findings

Complete characterization of extremal distributions in cycle scenarios with arbitrary outcomes.

Extension of methods to generalized scenarios beyond simple cycle gluings.

Application of homotopical and bundle techniques to contextuality analysis.

Abstract

Cycle scenarios are a significant class of contextuality scenarios, with the Clauser-Horne-Shimony-Holt (CHSH) 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.