Contextuality-by-default for behaviours in compatibility scenarios

TL;DR

This paper demonstrates that the core concept of contextuality-by-default is inherently present in the compatibility-hypergraph approach, introduces a non-degeneracy condition, and explores extended contextuality to deepen understanding of behaviors in contextuality scenarios.

Contribution

It shows the implicit presence of CbD ideas in CA, introduces a non-degeneracy condition, and develops the concept of extended contextuality for compatibility scenarios.

Findings

Non-degeneracy condition is weaker than non-disturbance.

Behaviors form a polytope allowing linear inequality characterization.

Non-degenerate behaviors are equivalent to standard non-contextual behaviors.

Abstract

We show that the main idea behind contextuality-by-default (CbD), i.e., the assumption that a physical measurement has to be understood as a contextual collection of random variables, is implicit in the compatibility-hypergraph approach to contextuality (CA) and use this result to develop in the latter important concepts which were introduced in the former. We introduce in CA the non-degeneracy condition, which is the analogous of consistent connectedness, and prove that this condition is, in general, weaker than non-disturbance condition. The set of non-degenerate behaviours defines a polytope, implying that one can characterize consistent connectedness using linear inequalities. We introduce the idea of extended contextuality for behaviours and prove that a behaviour is non-contextual in the standard sense iff it is non-degenerate and non-contextual in the extended sense. Finally, we use extended scenarios and behaviours to shed new light on our results.