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

Several principled measures of contextuality have been proposed for general systems of random variables (i.e. inconsistently connected systems). One such measure is based on quasi-couplings using negative probabilities (here denoted by [Formula: see text], Dzhafarov & Kujala, 2016 Quantum interaction). Dzhafarov & Kujala (Dzhafarov & Kujala 2019 Phil. Trans. R. Soc. A 377, 20190149. (doi:10.1098/rsta.2019.0149)) introduced a measure of contextuality, [Formula: see text], that naturally generalizes to a measure of non-contextuality. Dzhafarov & Kujala (Dzhafarov & Kujala 2019 Phil. Trans. R. Soc. A 377, 20190149. (doi:10.1098/rsta.2019.0149)) additionally conjectured that in the class of cyclic systems these two measures are proportional. Here we prove that conjecture is correct. Recently, Cervantes (Cervantes 2023 J. Math. Psychol. 112, 102726. (doi:10.1016/j.jmp.2022.102726)) showed the proportionality of [Formula: see text] and the Contextual Fraction measure introduced by Abramsky & Brandenburger (Abramsky & Brandenburger 2011 New J. Phys. 13, 113036. (doi:10.1088/1367-2630/13/11/113036)). The present proof completes the description of the interrelations of all contextuality measures proposed within or translated into the Contextuality-by-Default framework so far as they pertain to cyclic systems. This article is part of the theme issue 'Quantum contextuality, causality and freedom of choice'.

Contextuality Analysis of Impossible Figures.

This paper has two purposes. One is to demonstrate contextuality analysis of systems of epistemic random variables. The other is to evaluate the performance of a new, hierarchical version of the measure of (non)contextuality introduced in earlier publications. As objects of analysis we use impossible figures of the kind created by the Penroses and Escher. We make no assumptions as to how an impossible figure is perceived, taking it instead as a fixed physical object allowing one of several deterministic descriptions. Systems of epistemic random variables are obtained by probabilistically mixing these deterministic systems. This probabilistic mixture reflects our uncertainty or lack of knowledge rather than random variability in the frequentist sense.

Measures of contextuality and non-contextuality.

We discuss three measures of the degree of contextuality in contextual systems of dichotomous random variables. These measures are developed within the framework of the Contextuality-by-Default (CbD) theory, and apply to inconsistently connected systems (those with 'disturbance' allowed). For one of these measures of contextuality, presented here for the first time, we construct a corresponding measure of the degree of non-contextuality in non-contextual systems. The other two CbD-based measures do not suggest ways in which degree of non-contextuality of a non-contextual system can be quantified. We find the same to be true for the contextual fraction measure developed by Abramsky, Barbosa and Mansfield. This measure of contextuality is confined to consistently connected systems, but CbD allows one to generalize it to arbitrary systems. This article is part of the theme issue 'Contextuality and probability in quantum mechanics and beyond'.