Quantum graph isomorphism–strategy correspondence conjecture

Let $G:X\to X$ and $H:A\to A$ be irreflexive quantum graphs. The quantum graph isomorphism game is

$$\lambda_{G\simeq H}:X\otimes X^\mathrm{op}\to A\otimes A^\mathrm{op},$$

with

$$\lambda_{G\simeq H}:=\lambda_{G\to H}\star\lambda_{H\to G}^\dagger.$$

A quantum graph isomorphism from $G$ to $H$ is the corresponding structure-preserving quantum map. Quantum graph isomorphism–strategy correspondence conjecture. There is a one-to-one correspondence between quantum graph isomorphisms from $G$ to $H$ and quantum tensor bistrategies realizing perfect bicorrelations, with respect to a cup state, for the quantum graph isomorphism game $\lambda_{G\simeq H}$. The theorem preceding this conjecture establishes bisynchronicity, constructs perfect bicorrelations from quantum graph isomorphisms using quantum bistrategies and a cup state, and obtains quantum graph isomorphisms from perfect quantum commuting bistrategies; the full tensor-strategy correspondence remains unproved.