Quantum graphs of homomorphisms

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.