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

Abstract

Starting from arbitrary sets of quantum states and measurements, referred to as the prepare-and-measure scenario, an operationally noncontextual ontological model of the quantum statistics associated with the prepare-and-measure scenario is constructed. The operationally noncontextual ontological model coincides with standard Spekkens noncontextual ontological models for tomographically complete scenarios, while covering the non-tomographically complete case with a new notion of a reduced space, which we motivate following the guiding principles of noncontextuality. A mathematical criterion, called unit separability, is formulated as the relevant classicality criterion – the name is inspired by the usual notion of quantum state separability. Using this criterion, 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 of linear constraints, from which we obtain two separate hierarchies of algorithmic tests to witness the non-classicality or certify the classicality of a scenario. Finally, we reformulate our results in the framework of generalized probabilistic theories and discuss the implications for simplex-embeddability in such theories.

Accessible fragments of generalized probabilistic theories, cone equivalence, and applications to witnessing nonclassicality

“…Note that Gitton and Woods[65] introduced a notion of a "reduced space" based on quotienting relative to an equivalence relation wherein states are equivalent relative to the accessible effects and wherein effects are equivalent relative to the accessible states. This "reduced space" characterization of an experimental scenario is quite distinct from ours and their resulting notion of classicality is quite distinct from generalized noncontextuality.2 This latter point is similar to how in noncausal GPTs[3,66] there are states that do not have a normalized counterpart[3], but it occurs for a different reason, namely, the fact that a given experiment may not have access to repeat-until-success preparations.…”