Extending resource monotones as Kan extensions

Whats in these talks? Introduction to resource theories, introduction to Kan extensions, computing Kan extensions, partitioned categories (pCats), pCat functors, application of Kan extensions to extending monotones

This talk was presented at the 5th International Conference in Applied Category Theory 2022 based on this paper. A video recording of the talk can be found here. 

Abstract: In physics, a resource theory is used to model physical systems for which certain transformations are considered to be "free of cost". For example, on a hot summer day, cooling down the water requires energy (used by a refrigerator), hence is not free. However, a glass of chilled water warming up to room temperature is a free transformation. Resources are states of such a system. Monotones assigns a real number to each resource based on their value or utility. A problem in physics is that, can monotones on one resource theory be extended to another in case the two theories are related.

Gour and Tomamichel studied the problem of extending monotones using set-theoretical framework when a resource theory embeds fully and faithfully into the larger theory. One can generalize the problem of computing monotone extensions to scenarios when there exists a functorial transformation of one resource theory to another instead of just a full and faithful inclusion. In this talk, I show that (point-wise) Kan extensions provide a precise categorical framework to describe and compute such extensions of monotones. 

To set up monotone extensions using Kan extensions, we introduce partitioned categories (pCat) as a framework for resource theories and pCat functors to formalize relationship between resource theories. We describe monotones as pCat functors into ([0,∞],≤), and describe extending monotones along any pCat functor using Kan extensions. We show how our framework works by applying it to extend entanglement monotones for bipartite pure states to bipartite mixed states, to extend classical divergences to the quantum setting, and to extend a non-uniformity monotone from classical probabilistic theory to quantum theory.