Noncontextual ontological models of operational probabilistic theories

TL;DR

This paper extends the concept of noncontextual ontological models to operational probabilistic theories, proving that strongly causal theories admit noncontextual ontological representations independent of instrument structure.

Contribution

It adapts noncontextuality to OPTs with explicit instrument structures and proves noncontextuality for strongly causal quotiented OPTs, showing their representations depend only on transformations.

Findings

Noncontextual models can be adapted to OPTs with explicit instrument structures.

Strongly causal quotiented OPTs admit noncontextual ontological representations.

Ontological representations are determined solely by transformations, not instrument structure.

Abstract

An experiment or theory is classically explainable if it can be reproduced by some noncontextual ontological model. In this work, we adapt the notion of ontological models and generalized noncontextuality so it applies to the framework of operational probabilistic theories (OPTs). A defining feature of quotiented OPTs, which sets them apart from the closely related framework of generalized probabilistic theories (GPTs), is their explicit specification of the structure of instruments, these being generalizations of $\textit{quantum instruments}$ (including nondestructive measurements); in particular, one needs to explicitly declare which collections of transformations constitute a valid instrument. We are particularly interested in strongly causal OPTs, in which the choice of a future instrument can be conditioned on a past measurement outcome. This instrument structure might seem to permit the possibility of a contextual kind of ontological representation, where the representation of a given transformation depends on which instrument it is considered a part of. However, we prove that this is not possible by showing that for strongly causal quotiented OPTs the structure of instruments does $\textit{not}$ allow for such a contextual ontological representation. It follows that ontological representations of strongly causal quotiented OPTs are entirely determined by their action on individual transformations, with no dependence on the structure of instruments.