Suboptimality of parity for distilling correlations with nontrivial marginals

We prove that the parity protocol is optimal for a general class of nonadaptive distillation protocols of all $n$-player nonlocal boxes (NLBs) based on xor games. The conditional distributions generated by these NLBs are assumed to have trivial local marginals. We also show that already for $n=2$, parity is no longer optimal if the local marginals are nontrivial. The or protocol is shown to perform better and in the process also slightly extend the known correlations that collapse communication complexity. This emphasizes again the need to understand the local properties of nonlocal systems in order to obtain a better characterization of the global behavior. We conclude by showing an equivalence between adaptive distillation protocols that use identical NLBs and parity protocol using nonidentical NLBs.