# A Profunctorial Semantics for Quantum Supermaps

arXiv: https://arxiv.org/abs/2402.02997 · HTML: https://mlsanity.com/paper/2402.02997/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 5 Feb 2024.

## Built on by (most cited first, 1 of 1)

- [Agent policies from higher-order causal functions](https://mlsanity.com/paper/2512.10937/) (2025)

## Abstract (arXiv)

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.