A structure theorem for generalized-noncontextual ontological models

TL;DR

This paper extends the framework of generalized noncontextuality to arbitrary compositional scenarios, proving a structural theorem that constrains ontological models and links classical explanations to quasiprobability representations.

Contribution

It provides a formal extension of ontological models and generalized noncontextuality to complex scenarios, revealing a rigid structure and establishing equivalences among classicality notions.

Findings

Every noncontextual ontological model has a simple, non-overcomplete frame representation.

The maximum number of ontic states equals the dimension of the associated GPT.

The results facilitate noncontextuality no-go theorems and experimental certification of contextuality.

Abstract

It is useful to have a criterion for when the predictions of an operational theory should be considered classically explainable. Here we take the criterion to be that the theory admits of a generalized-noncontextual ontological model. Existing works on generalized noncontextuality have focused on experimental scenarios having a simple structure: typically, prepare-measure scenarios. Here, we formally extend the framework of ontological models as well as the principle of generalized noncontextuality to arbitrary compositional scenarios. We leverage a process-theoretic framework to prove that, under some reasonable assumptions, every generalized-noncontextual ontological model of a tomographically local operational theory has a surprisingly rigid and simple mathematical structure -- in short, it corresponds to a frame representation which is not overcomplete. One consequence of this theorem is that the largest number of ontic states possible in any such model is given by the dimension of the associated generalized probabilistic theory. This constraint is useful for generating noncontextuality no-go theorems as well as techniques for experimentally certifying contextuality. Along the way, we extend known results concerning the equivalence of different notions of classicality from prepare-measure scenarios to arbitrary compositional scenarios. Specifically, we prove a correspondence between the following three notions of classical explainability of an operational theory: (i) existence of a noncontextual ontological model for it, (ii) existence of a positive quasiprobability representation for the generalized probabilistic theory it defines, and (iii) existence of an ontological model for the generalized probabilistic theory it defines.