We propose a framework, named the postselected inflation framework, to obtain converging outer approximations of the sets of probability distributions that are compatible with classical multi-network scenarios. Here, a network is a bilayer directed acyclic graph with a layer of sources of classical randomness, a layer of agents, and edges specifying the connectivity between the agents and the sources. A multi-network scenario is a list of such networks, together with a specification of subsets of agents using the same strategy. We furthermore show that the postselected inflation framework is mathematically equivalent to the standard inflation framework: in that respect, our results allow to gain further insights into the convergence proof of the inflation hierarchy of Navascuès and Wolfe, and extend it to the case of multi-network scenarios.
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.
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