Contextuality of general probabilistic theories

TL;DR

This paper characterizes the relationship between generalized contextuality and general probabilistic theories (GPTs), revealing fundamental incompatibilities and conditions under which GPTs are ontologically noncontextual or contextual.

Contribution

It provides a comprehensive characterization of GPTs regarding their compatibility with noncontextual ontological models, based on a Gleason-type theorem and the no-restriction hypothesis.

Findings

GPTs cannot satisfy all three conditions simultaneously: no-restriction, noncontextuality, and multiple measurements.

GPTs satisfying the no-restriction hypothesis are noncontextual only if they are simple.

SubGPTs are noncontextual if they are subtheories of simplicial GPTs of the same dimension.

Abstract

Generalized contextuality refers to our inability of explaining measurement statistics using a context-independent probabilistic and ontological model. On the other hand, measurement statistics can also be modeled using the framework of general probabilistic theories (GPTs). Here, starting from a construction of GPTs based on a Gleason-type theorem, we fully characterize these structures with respect to their permission and rejection of generalized (non)contextual ontological models. It follows that in any GPT construction the three insistence of (i) the no-restriction hypothesis, (ii) the ontological noncontextuality, and (iii) multiple nonrefinable measurements for any fixed number of outcomes are incompatible. Hence, any GPT satisfying the no-restriction hypothesis is ontologically noncontextual if and only if it is simplicity. We give a detailed discussion of GPTs for which the no-restriction hypothesis is violated, and show that they can always be considered as subtheories (subGPTs) of GPTs satisfying the hypothesis. It is shown that subGPTs are ontologically noncontextual if and only if they are subtheories of simplicial GPTs of the same dimensionality. Finally, we establish as a corollary the necessary and sufficient condition for a single resourceful measurement or state to promote an ontologically noncontextual (i.e. classical) general probabilistic theory to an ontologically contextual (i.e. nonclassical) one under the no-restriction hypothesis.