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It is useful to have a criterion for when the predictions of an operationaltheory should be considered classically explainable. Here we take the criterionto be that the theory admits of a generalized-noncontextual ontological model.Existing works on generalized noncontextuality have focused on experimentalscenarios having a simple structure, typically, prepare-measure scenarios.Here, we formally extend the framework of ontological models as well as theprinciple of generalized noncontextuality to arbitrary compositional scenarios.We leverage this process-theoretic framework to prove that, under somereasonable assumptions, every generalized-noncontextual ontological model of atomographically local operational theory has a surprisingly rigid and simplemathematical structure; in short, it corresponds to a frame representationwhich is not overcomplete. One consequence of this theorem is that the largestnumber of ontic states possible in any such model is given by the dimension ofthe associated generalized probabilistic theory. This constraint is useful forgenerating noncontextuality no-go theorems as well as techniques forexperimentally certifying contextuality. Along the way, we extend known resultsconcerning the equivalence of different notions of classicality fromprepare-measure scenarios to arbitrary compositional scenarios. Specifically,we prove a correspondence between the following three notions of classicalexplainability of an operational theory: (i) admitting a noncontextualontological model, (ii) admitting of a positive quasiprobabilityrepresentation, and (iii) being simplex-embeddable.

A structure theorem for generalized-noncontextual ontological models