Closing Bell: Boxing black box simulations in the resource theory of contextuality
Rui Soares Barbosa, Martti Karvonen, Shane Mansfield
TL;DR
The chapter develops a unified, resource-theoretic view of contextuality using the sheaf-theoretic framework, recasting contextual behaviours as empirical models and transformations between them as simulations. Central to the approach is the internal hom construction $[S,T]$, which internalizes morphisms and yields a closed category structure when paired with structure predicates $g_{S,T}$; a map $F: ext{EMP}(S) o ext{EMP}(T)$ is realizable by a procedure $S o T$ precisely when $F$ is induced by a non-contextual model on $[S,T]$. This yields a concrete criterion that connects free classical simulations to non-contextuality, unifying non-local games and contextuality within the same operational framework. The work also analyzes variants (possibilistic, adaptive, structured-predicate) and outlines future directions, including links to cohomology and dual algebraic formulations, with broad implications for understanding contextuality as a resource in computation and information processing.
Abstract
This chapter contains an exposition of the sheaf-theoretic framework for contextuality emphasising resource-theoretic aspects, as well as some original results on this topic. In particular, we consider functions that transform empirical models on a scenario S to empirical models on another scenario T, and characterise those that are induced by classical procedures between S and T corresponding to 'free' operations in the (non-adaptive) resource theory of contextuality. We construct a new 'hom' scenario built from S and T, whose empirical models induce such functions. Our characterisation then boils down to being induced by a non-contextual model. We also show that this construction on scenarios provides a closed structure on the category of measurement scenarios.