# Contextuality and Noncontextuality Measures and Generalized Bell Inequalities for Cyclic Systems

## TL;DR

This paper develops new measures of contextuality and noncontextuality for cyclic systems of dichotomous variables within the CbD framework, extending analysis to inconsistently connected systems and providing a complete geometric characterization.

## Contribution

It introduces novel L1-distance based measures for (non)contextuality applicable to all cyclic systems, including inconsistently connected ones, and characterizes the associated polytopes.

## Key findings

- Complete geometric characterization of noncontextuality polytope.
- Extension of measures to inconsistently connected systems.
- Theoretical analysis of contextuality degrees in cyclic systems.

## Abstract

Cyclic systems of dichotomous random variables have played a prominent role in contextuality research, describing such experimental paradigms as the Klyachko-Can-Binicoglu-Shumovky, Einstein-Podolsky-Rosen-Bell, and Leggett-Garg ones in physics, as well as conjoint binary choices in human decision making. Here, we understand contextuality within the framework of the Contextuality-by-Default (CbD) theory, based on the notion of probabilistic couplings satisfying certain constraints. CbD allows us to drop the commonly made assumption that systems of random variables are consistently connected. Consistently connected systems constitute a special case in which CbD essentially reduces to the conventional understanding of contextuality. We present a theoretical analysis of the degree of contextuality in cyclic systems (if they are contextual) and the degree of noncontextuality in them (if they are not). By contrast, all previously proposed measures of contextuality are confined to consistently connected systems, and most of them cannot be extended to measures of noncontextuality. Our measures of (non)contextuality are defined by the L_{1}-distance between a point representing a cyclic system and the surface of the polytope representing all possible noncontextual cyclic systems with the same single-variable marginals. We completely characterize this polytope, as well as the polytope of all possible probabilistic couplings for cyclic systems with given single-variable marginals.[...]