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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. We analyse dualisable and invertible $1$-morphisms in these $2$-categories and show that they correspond precisely to the existing notions of quantum isomorphism and classical isomorphism between sets and graphs.'
bibliography:
- 'QSET.bib'
title: A compositional approach to quantum functions
---

Introduction
============

#### On quantum functions.

In this work, we introduce a notion of *quantum function* between finite *quantum sets* and propose a framework for finite quantum set theory.

These quantum functions underlie various notions of ‘quantum morphism’ previously defined in quantum information theory and noncommutative topology.[^1]

Several such notions

have recently emerged from the study of *quantum pseudo-telepathy* [@Brassard2005], a phenomenon in quantum theory where pre-shared entanglement is 