On Joint Distributions, Counterfactual Values, and Hidden Variables in Relation to Contextuality

TL;DR

This paper explores the relationships between different definitions of contextuality in quantum systems, demonstrating that within the Contextuality-by-Default framework, joint distributions, counterfactuals, and hidden variables are interconnected through probabilistic couplings.

Contribution

It shows that the CbD framework unifies various notions of contextuality by embedding counterfactuals and hidden variables into the concept of probabilistic couplings.

Findings

Probabilistic couplings unify different contextuality definitions.

Hidden variables can be viewed as specific couplings.

C1, C2, and C3 notions of contextuality are interconnected within CbD.

Abstract

This paper deals with three traditional ways of defining contextuality: (C1) in terms of (non)existence of certain joint distributions involving measurements made in several mutually exclusive contexts; (C2) in terms of relationship between factual measurements in a given context and counterfactual measurements that could be made if one used other contexts; and (C3) in terms of (non)existence of ``hidden variables'' that determine the outcomes of all factually performed measurements. It is generally believed that the three meanings are equivalent, but the issues involved are not entirely transparent. Thus, arguments have been offered that C2 may have nothing to do with C1, and the traditional formulation of C1 itself encounters difficulties when measurement outcomes in a contextual system are treated as random variables. I show that if C1 is formulated within the framework of the Contextuality-by-Default (CbD) theory, the notion of a probabilistic coupling, the core mathematical tool of CbD, subsumes both counterfactual values and ``hidden variables''. In the latter case, a coupling itself can be viewed as a maximally parsimonious choice of a hidden variable.