Quantum graphs of homomorphisms

• URL: http://arxiv.org/abs/2601.09685v1
• Date: Wed, 14 Jan 2026 18:36:43 GMT
• Title: Quantum graphs of homomorphisms
• Authors: Andre Kornell, Bert Lindenhovius,
• Abstract summary: category $mathsfqGph$ of quantum graphs.<n> quantum graph $[G,H]$ is nonempty iff the $(G,H)$-homomorphism game has a winning quantum strategy.<n>Every finite reflexive quantum graph is the confusability quantum graph of a quantum channel.
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• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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.

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