Reassessing the boundary between classical and nonclassical for individual quantum processes

There is a received wisdom about where to draw the boundary between classical and nonclassical for various types of quantum processes. For multipartite states, it is the divide between separable and entangled; for channels, the divide between entanglement-breaking and not; for sets of measurements, the divide between compatible and incompatible; for assemblages, the divide between steerable and unsteerable. However, these choices have not been motivated by any unified notion of what it means to be classically explainable. One well-motivated notion of classical explainability is the one based on generalized noncontextuality: a set of circuits is classically explainable if a generalized-noncontextual ontological model can realize the statistics they generate. In this work, we show that this notion can be leveraged to define a classical-nonclassical divide for individual quantum processes of arbitrary type. We begin the task of characterizing where the classical-nonclassical divide lies according to this proposal for a variety of different types of processes. In particular, we show that all of the following are judged to be nonclassical: every entangled state, every set of incompatible measurements, every non-entanglement-breaking channel, and every steerable assemblage. Our proposal differs from the received wisdom, however, insofar as it also judges certain subsets of the complementary classes to be nonclassical, including certain separable states, compatible sets of measurements, entanglement-breaking channels, and unsteerable assemblages. Finally, we prove structure theorems characterizing the classical-nonclassical divide based on whether a process admits of a specific type of frame representation.