Quantum graphs of homomorphisms
Andre Kornell, Bert Lindenhovius
TL;DR
The paper constructs the category $\mathsf{qGph}$ of quantum graphs from noncommutative geometry and defines a quantum graph $[G,H]$ of homomorphisms for each pair of quantum graphs. It proves that $\mathsf{qGph}$ is closed and that $[G,H]$ satisfies a universal property analogous to evaluation, with a crucial equivalence: finite graphs $G,H$ admit a winning quantum strategy in the $(G,H)$-homomorphism game iff $[G,H]$ is nonempty, linking quantum homomorphisms to nonlocal games. The approach unifies quantum function spaces (via quantum sets and relations) with quantum strategies for nonlocal games, and shows that morphisms in $\mathsf{qGph}$ correspond to trace-nonincreasing CP maps adjoint to unital $*$-homomorphisms, interpreting reflexive quantum relations as confusability graphs of quantum channels. The results illuminate a deep interaction between noncommutative geometry, quantum information, and graph homomorphisms, and provide a framework where classical and quantum homomorphisms are treated uniformly within a closed monoidal setting.
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 show that Weaver's two notions of a CP morphism coincide in this context. We also show that every finite reflexive quantum graph is the confusability quantum graph of a quantum channel, answering a question of Daws.