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

Abstract

The formalism of generalized probabilistic theories (GPTs) was originally developed as a way to characterize the landscape of conceivable physical theories. Thus, the GPT describing a given physical theory necessarily includes all physically possible processes. We here consider the question of how to provide a GPT-like characterization of a particular experimental setup within a given physical theory. We show that the resulting characterization is not generally a GPT in and of itself-rather, it is described by a more general mathematical object that we introduce and term an accessible GPT fragment. We then introduce an equivalence relation, termed cone equivalence, between accessible GPT fragments (and, as a special case, between standard GPTs). We give a number of examples of experimental scenarios that are best described using accessible GPT fragments, and where moreover cone-equivalence arises naturally. We then prove that an accessible GPT fragment admits of a classical explanation if and only if every other fragment that is cone-equivalent to it also admits of a classical explanation. Finally, we leverage this result to prove several fundamental results regarding the experimental requirements for witnessing the failure of generalized noncontextuality. In particular, we prove that neither incompatibility among measurements nor the assumption of freedom of choice is necessary for witnessing failures of generalized noncontextuality, and, moreover, that such failures can be witnessed even using arbitrarily inefficient detectors.

An open-source linear program for testing nonclassicality

“…Note that satisfiability is only a function of the cones defined by Ω and by E, and so no other features of the states and effects are relevant to their nonclassicality, as was also shown in Ref. [2,34]. A useful consequence of this fact is that Ω and E are classically-explainable if and only if their convex hulls are also classically-explainable.…”

“…Using the flag-convexification technique of Ref. [2,34], we suspect that all prepare-measure scenarios can be transformed into prepare-measure scenarios of this particular type, in which case the linear program from Ref. [30] would be as general as the approach we have discussed herein.…”

“…Ref. [2] contains a comprehensive introduction to the particular formalization of GPTs which we use here. In addition, see Refs.…”

“…In Refs. [1,2], it was shown that the appropriate notion of classical-explainability is the notion of simplex-embeddability. This geometric criterion deems a GPT classically-explainable if its state space can be embedded into a simplex (of any dimension) and its effect space can be embedded into the dual to the simplex, such that probabilities are preserved.…”

“…A large number of previous works have studied the question of when a set of data admits of a generalized noncontextual model [1,2,6,7,[30][31][32][33][34][35]. Most closely related to our work are Refs.…”