A Profunctorial Semantics for Quantum Supermaps
James Hefford, Matt Wilson
TL;DR
The paper develops a theory-agnostic categorical model for quantum supermaps by treating them as morphisms of strong profunctors, situated in copresheaf semantics over optics. It identifies two interacting tensor products, $-\otimes_\mathcal{C}-$ and $-\varogreaterthan-$, enriching the category $\mathsf{StProf}(\mathcal{C})$ to a normal duoidal structure and enabling a unified treatment of indefinite and definite causal order through representability and Yoneda embeddings. The authors establish decomposition theorems and duality principles, connect locality laws to optic/combo formalisms, and introduce a tensorial logic (dialogue categories) to reason about higher-order maps without requiring full $*$-autonomy. The framework recovers known CPTP results via optics/combs equivalence, extends to multi-party settings with semi-localisable and separable spaces, and provides a solid foundation for future work on quantum foundations and beyond, including OPTs and more general theories.
Abstract
We identify morphisms of strong profunctors as a categorification of quantum supermaps. These black-box generalisations of diagrams-with-holes are hence placed within the broader field of profunctor optics, as morphisms in the category of copresheaves on concrete networks. This enables the first construction of abstract logical connectives such as tensor products and negations for supermaps in a totally theory-independent setting. These logical connectives are found to be all that is needed to abstractly model the key structural features of the quantum theory of supermaps: black-box indefinite causal order, black-box definite causal order, and the factorisation of definitely causally ordered supermaps into concrete circuit diagrams. We demonstrate that at the heart of these factorisation theorems lies the Yoneda lemma and the notion of representability.