Suboptimality of Parity for Distilling Correlations with Nontrivial Marginals

TL;DR

This paper analyzes the optimality of the PARITY protocol in distilling correlations from nonlocal boxes, revealing its limitations with non-trivial marginals and proposing the OR protocol as a better alternative.

Contribution

It proves PARITY's optimality for certain non-adaptive protocols with trivial marginals and demonstrates its suboptimality with non-trivial marginals, introducing the OR protocol as an improvement.

Findings

PARITY is optimal for non-adaptive protocols with trivial marginals.

PARITY is not optimal when local marginals are non-trivial.

The OR protocol outperforms PARITY in certain correlation distillation scenarios.

Abstract

We prove that the PARITY protocol is optimal for a general class of non-adaptive 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 non-trivial. 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.