Quantum graphs of homomorphisms
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2026
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| _version_ | 1866917444424040448 |
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| author | Kornell, Andre Lindenhovius, Bert |
| author_facet | Kornell, Andre Lindenhovius, Bert |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_09685 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Quantum graphs of homomorphisms Kornell, Andre Lindenhovius, Bert Quantum Physics Mathematical Physics Category Theory Operator Algebras 46L89, 05C76, 81P40, 81P47, 18D15 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. |
| title | Quantum graphs of homomorphisms |
| topic | Quantum Physics Mathematical Physics Category Theory Operator Algebras 46L89, 05C76, 81P40, 81P47, 18D15 |
| url | https://arxiv.org/abs/2601.09685 |