A comonadic view of simulation and quantum resources

TL;DR

This paper presents a simplified and generalized categorical framework for understanding simulation and quantum resources within the sheaf-theoretic approach to contextuality and non-locality, emphasizing morphisms and resource theories.

Contribution

It introduces a streamlined notion of morphism for empirical models and extends it via a co-Kleisli category, capturing simulation notions in quantum resource theories.

Findings

Morphisms capture simulation between empirical models.

The framework applies to measurement-based quantum computation.

Simplifies previous approaches to resource theories.

Abstract

We study simulation and quantum resources in the setting of the sheaf-theoretic approach to contextuality and non-locality. Resources are viewed behaviourally, as empirical models. In earlier work, a notion of morphism for these empirical models was proposed and studied. We generalize and simplify the earlier approach, by starting with a very simple notion of morphism, and then extending it to a more useful one by passing to a co-Kleisli category with respect to a comonad of measurement protocols. We show that these morphisms capture notions of simulation between empirical models obtained via `free' operations in a resource theory of contextuality, including the type of classical control used in measurement-based quantum computation schemes.