Measures of contextuality in cyclic systems and the negative probabilities measure CNT3

TL;DR

This paper proves that two measures of contextuality, CNT3 and CNT2, are proportional in cyclic systems, completing the understanding of their interrelations and advancing the theoretical framework of contextuality measures.

Contribution

It provides a proof that CNT3 and CNT2 are proportional in cyclic systems, clarifying their relationship and completing the framework of contextuality measures.

Findings

CNT3 and CNT2 are proportional in cyclic systems

The proof completes the interrelation map of contextuality measures

Advances the theoretical understanding of contextuality in cyclic systems

Abstract

Several principled measures of contextuality have been proposed for general systems of random variables (i.e. inconsistentlly connected systems). The first of such measures was based on quasi-couplings using negative probabilities (here denoted by CNT3, Dzhafarov & Kujala, 2016). Dzhafarov and Kujala (2019) introduced a measure of contextuality, CNT2, that naturally generalizes to a measure of non-contextuality. Dzhafarov and Kujala (2019) additionally conjectured that in the class of cyclic systems these two measures are proportional. Here we prove that that conjecture is correct. Recently, Cervantes (2023) showed the proportionality of CNT2 and the Contextual Fraction measure (CNTF) introduced by Abramsky, Barbosa, and Mansfeld (2017). The present proof completes the description of the interrelations of all contextuality measures as they pertain to cyclic systems.