  To make precise the sense in which the operational predictions of quantum
theory conflict with a classical worldview, it is necessary to articulate a
notion of classicality within an operational framework. A widely applicable
notion of classicality of this sort is whether or not the predictions of a
given operational theory can be explained by a generalized-noncontextual
ontological model. We here explore what notion of classicality this implies for
the generalized probabilistic theory (GPT) that arises from a given operational
theory, focusing on prepare-measure scenarios. We first show that, when mapping
an operational theory to a GPT by quotienting relative to operational
equivalences, the constraint of explainability by a generalized-noncontextual
ontological model is mapped to the constraint of explainability by an
ontological model. We then show that, under the additional assumption that the
ontic state space is of finite cardinality, this constraint on the GPT can be
expressed as a geometric condition which we term simplex-embeddability. Whereas
the traditional notion of classicality for a GPT is that its state space be a
simplex and its effect space be the dual of this simplex, simplex-embeddability
merely requires that its state space be embeddable in a simplex and its effect
space in the dual of that simplex. We argue that simplex-embeddability
constitutes an intuitive and freestanding notion of classicality for GPTs. Our
result also has applications to witnessing nonclassicality in prepare-measure
experiments.
