A comonadic view of simulation and quantum resources

Samson Abramsky (University of Oxford), Rui Soares Barbosa (University of Oxford), Martti Karvonen (University of Edinburgh), Shane Mansfield (LIP6, Sorbonne Université)

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 a 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.