Quantum graphs of homomorphisms

TL;DR

This paper develops a category of quantum graphs inspired by noncommutative geometry, constructs a homomorphism quantum graph, and relates it to quantum strategies in graph homomorphism games.

Contribution

It introduces a new categorical framework for quantum graphs, generalizes classical homomorphism concepts, and connects quantum graph properties with quantum strategies.

Findings

Quantum graph $[G,H]$ exists iff the $(G,H)$-homomorphism game has a quantum winning strategy.

Finite quantum graphs are tracial, real, self-adjoint, with CP morphisms coinciding with unital *-homomorphisms.

Every finite reflexive quantum graph is a confusability graph of a quantum channel.

Abstract

We introduce a category $\mathsf{qGph}$ of quantum graphs, whose definition is motivated entirely from noncommutative geometry. For all quantum graphs $G$ and $H$ in $\mathsf{qGph}$, we then construct a quantum graph $[G,H]$ of homomorphisms from $G$ to $H$, making $\mathsf{qGph}$ a closed symmetric monoidal category. We prove that for all finite graphs $G$ and $H$, the quantum graph $[G,H]$ is nonempty iff the $(G,H)$-homomorphism game has a winning quantum strategy, directly generalizing the classical case. The finite quantum graphs in $\mathsf{qGph}$ are tracial, real, and self-adjoint, and the morphisms between them are CP morphisms that are adjoint to a unital $*$-homomorphism. We prove that Weaver's two notions of a CP morphism coincide in this context. We also include a short proof that every finite reflexive quantum graph is the confusability quantum graph of a quantum channel.