Supermaps on generalised theories
This work provides a foundational mathematical tool for defining higher-order processes in arbitrary physical theories, eliminating ambiguity for researchers working on generalised probabilistic theories.
The authors prove a Yoneda lemma for categorical supermaps, showing that in theories with channel-state duality, supermaps can be concretely represented via that duality. This provides a unified framework for defining higher-order processes in generalised probabilistic theories, with applications to boxworld and real quantum theory.
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.