Convexity of noncontextual wirings and how they order the set of correlations

TL;DR

This paper explores the structure and properties of noncontextual wirings in the resource theory of contextuality, demonstrating their convexity, their relation to local operations, and their role in ordering correlations.

Contribution

It establishes the convexity of noncontextual wirings, compares them to LOSR operations, and analyzes resource conversion and ordering in contextuality scenarios.

Findings

Noncontextual wirings form a convex set.

NCW operations are larger than LOSR in Bell scenarios.

Not all behaviors can be converted via NCW.

Abstract

The resource theory of contextuality considers resourceful objects to be probabilistic data-tables, known as correlations or behaviors, that fail to have an explanation in terms of Kochen-Specker noncontextual models. In this work, we advance this resource theory, considering free operations to be noncontextual wirings (NCW). We show that all such wirings form a convex set. When restricted to Bell scenarios, we show that such wirings are not equivalent to local operations assisted by a common source of classical shared randomness (LOSR). The set of all NCW operations contains LOSR, but is strictly larger. We also prove several elementary facts about how different resources can be converted via NCW. As a concrete example, we show that there are pairs of behaviors that cannot be converted one into the other using NCW. Since resource conversion mathematically induces a pre-order over the set of all behaviors, our results reveal the intricate ordering induced by NCW in scenarios beyond Bell scenarios.