Supermaps on generalised theories

TL;DR

This paper extends the concept of supermaps to generalised theories using category theory, establishing a foundational lemma and applying it to quantum and boxworld theories.

Contribution

It proves the Yoneda lemma for categorical supermaps, enabling concrete representations based on channel-state duality across various theories.

Findings

Yoneda lemma for categorical supermaps established

Higher-order processes on boxworld derived as a special case

Stable definition of higher-order real quantum theory proposed

Abstract

Categorical supermaps generalise higher-order quantum operations from finite-dimensional quantum theory to arbitrary circuit theories. In this paper, we establish the Yoneda lemma for categorical supermaps, which states that whenever a physical theory has a suitable notion of channel-state duality, then categorical supermaps on that theory can be concretely represented in terms of that duality. This lemma eliminates any guesswork or ambiguity when defining the appropriate notion of supermap for these theories. As a concrete application, we show that the recently proposed higher-order processes on boxworld can be obtained as a particular instance of categorical supermaps, and put forward a stable definition of higher-order real quantum theory.