Quantum graph homomorphism–strategy correspondence conjecture

Let $G:X\to X$ and $H:A\to A$ be quantum graphs. A quantum homomorphism from $G$ to $H$ is a quantum function $E:X\to_{\mathcal K}A$ satisfying the quantum graph homomorphism condition, and $\lambda_{G\to H}$ denotes the quantum graph homomorphism game. Quantum graph homomorphism–strategy correspondence conjecture. There is a one-to-one correspondence between quantum graph homomorphisms from $G$ to $H$ and quantum tensor strategies realizing perfect correlations, with respect to a cup state, for $\lambda_{G\to H}$. This would identify the existing notion of quantum graph homomorphism with perfect tensor strategies for the associated game; the paper proves partial results but does not establish the full correspondence.