Solvable Criterion for the Contextuality of Any Prepare-and-Measure Scenario

Starting from arbitrary sets of quantum states and measurements, referred toas the prepare-and-measure scenario, an operationally noncontextual ontologicalmodel of the quantum statistics associated with the prepare-and-measurescenario is constructed. The operationally noncontextual ontological modelcoincides with standard Spekkens noncontextual ontological models fortomographically complete scenarios, while covering the non-tomographicallycomplete case with a new notion of a reduced space, which we motivate followingthe guiding principles of noncontextuality. A mathematical criterion, calledunit separability, is formulated as the relevant classicality criterion – thename is inspired by the usual notion of quantum state separability. Using thiscriterion, we derive a new upper bound on the cardinality of the ontic space.Then, we recast the unit separability criterion as a (possibly infinite) set oflinear constraints, from which we obtain two separate hierarchies ofalgorithmic tests to witness the non-classicality or certify the classicalityof a scenario. Finally, we reformulate our results in the framework ofgeneralized probabilistic theories and discuss the implications forsimplex-embeddability in such theories.