A compositional approach to quantum functions

TL;DR

This paper develops a new compositional framework for finite quantum set theory using 2-categories, enabling a unified approach to quantum graphs, homomorphisms, and automorphism groups relevant to quantum information.

Contribution

It introduces quantum functions and a 2-categorical structure for quantum sets and graphs, linking quantum graph theory with quantum automorphism groups.

Findings

Quantum functions form a foundational notion for quantum set theory.

The 2-categories of quantum sets and graphs are semisimple.

Quantum permutations and isomorphisms are characterized as dagger-dualisable 1-morphisms.

Abstract

We introduce a notion of quantum function, and develop a compositional framework for finite quantum set theory based on a 2-category of quantum sets and quantum functions. We use this framework to formulate a 2-categorical theory of quantum graphs, which captures the quantum graphs and quantum graph homomorphisms recently discovered in the study of nonlocal games and zero-error communication, and relates them to quantum automorphism groups of graphs considered in the setting of compact quantum groups. We show that the 2-categories of quantum sets and quantum graphs are semisimple and characterise existing notions of quantum permutations and quantum graph isomorphisms as dagger-dualisable 1-morphisms in these 2-categories.