Quantum advantage in zero-error function computation with side information

We consider the problem of zero-error function computation with side information. Alice and Bob have correlated sources $X,Y$ with joint p.m.f. $p_{XY}(\cdot, \cdot)$. Bob wants to calculate $f(X,Y)$ with zero error. Alice communicates $m$-length blocks $(m \geq 1)$ with Bob over error-free channels: classical or quantum. In the classical setting, the minimum communication rate depends on the asymptotic growth of the chromatic number of an appropriately defined $m$-instance ``confusion graph'' $G^{(m)}$. In the quantum setting, it depends on the asymptotic growth of the orthogonal rank of the complement of $G^{(m)}$. The behavior of the quantum advantage (ratio of classical and quantum rates) depends critically on $G^{(m)}$ which is shown to be sandwiched between $G^{\boxtimes m}$ ($m$-times strong product) and $G^{\lor m}$ ($m$-times OR product) respectively. Our work presents necessary and sufficient conditions on the function $f(\cdot, \cdot)$ and joint p.m.f. $p_{XY}(\cdot,\cdot)$ such that $G^{(m)}$ equals either $G^{\boxtimes m}$ or $G^{\lor m}$. We study the behavior of the quantum advantage in the single-instance case and the asymptotic (in $m$) case for several classes of confusion graphs and demonstrate, e.g., that there exist problems where there is a quantum advantage in the single-instance rate but no quantum advantage in the asymptotic rate.
Comment: Added more examples for graph strong/OR products and for different types of quantum-advantages