Resource-theoretic hierarchy of contextuality for general probabilistic theories

TL;DR

This paper introduces a hierarchy of generalized contextuality within GPTs, defining a resource theory framework, a new monotone called classical excess, and analyzing operational measures like the parity oblivious multiplexing game.

Contribution

It develops a resource-theoretic hierarchy for contextuality in GPTs, introducing new monotones and operational measures to compare levels of contextuality.

Findings

Noncontextual theories are the least elements in the hierarchy.

The classical excess monotone quantifies minimal classical embedding error.

Success probability in the parity oblivious multiplexing game is a monotone.

Abstract

In this work we present a hierarchy of generalized contextuality. It refines the traditional binary distinction between contextual and noncontextual theories, and facilitates their comparison based on how contextual they are. Our approach focuses on the contextuality of prepare-and-measure scenarios, described by general probabilistic theories (GPTs). To motivate the hierarchy, we define it as the resource ordering of a novel resource theory of GPT-contextuality. The building blocks of its free operations are classical systems and univalent simulations between GPTs. These simulations preserve operational equivalences and thus cannot generate contextuality. Noncontextual theories can be recovered as least elements in the hierarchy. We then define a new contextuality monotone, called classical excess, given by the minimal error of embedding a GPT within an infinite classical system. In addition, we show that the optimal success probability in the parity oblivious multiplexing game also defines a monotone in our resource theory. Finally, we discuss whether the non-free operations can be understood as implementing information erasure and thus explaining the fine-tuning aspect of contextuality.