Varieties of contextuality based on probability and structural nonembeddability

TL;DR

This paper explores different types of contextuality in quantum mechanics, distinguishing probabilistic from strong forms using mathematical theorems, and relates them to classical and nonclassical models.

Contribution

It provides a unified framework to differentiate probabilistic and strong contextuality using probability theory and structural nonembeddability, based on Kochen-Specker's Theorem.

Findings

Probabilistic contextuality allows classical models with nonclassical probabilities.

Strong contextuality characterizes quantum observables with no Boolean algebra embedding.

Kochen-Specker's Theorem serves as a criterion to distinguish the two forms.

Abstract

Different analytic notions of contextuality fall into two major groups: probabilistic and strong notions of contextuality. Kochen and Specker's Theorem~0 is a demarcation criterion for differentiating between those groups. Whereas probabilistic contextuality still allows classical models, albeit with nonclassical probabilities, the logico-algebraic "strong" form of contextuality characterizes collections of quantum observables that have no faithfully embedding into (extended) Boolean algebras. Both forms indicate a classical in- or under-determination that can be termed "value indefinite" and formalized by partial functions of theoretical computer sciences.