TY - UNPB

T1 - A structure theorem for generalized-noncontextual ontological models

AU - Schmid, David

AU - Selby, John

AU - Pusey, Matthew Fairbairn

AU - Spekkens, Robert W.

PY - 2020/5/14

Y1 - 2020/5/14

N2 - 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 this 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) admitting a noncontextual ontological model, (ii) admitting of a positive quasiprobability representation, and (iii) being simplex-embeddable.

AB - 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 this 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) admitting a noncontextual ontological model, (ii) admitting of a positive quasiprobability representation, and (iii) being simplex-embeddable.

M3 - Preprint

BT - A structure theorem for generalized-noncontextual ontological models

ER -