Suboptimality of Parity for Distilling Correlations with Nontrivial Marginals

TL;DR

The paper analyzes distillation of weak multipartite nonlocal correlations in XOR games, showing that the PARITY protocol is optimal among non-adaptive distillation protocols with identical inputs when local marginals are trivial. It also demonstrates that non-trivial local marginals can enable the OR protocol to outperform PARITY in certain regimes, expanding the distillation landscape. A key result is a general equivalence between adaptive protocols with identical NLBs and non-adaptive PARITY protocols using non-identical NLBs, linking adaptive and non-adaptive strategies. The work emphasizes the importance of local marginal properties in shaping global nonlocal behavior and discusses implications for device-independent QKD and future characterization of distillation with non-trivial marginals.

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.

Figures (7)

• Figure 1: A multipartite NLBN for XOR games is shared between $n$ players. On input bit string $x$ the box outputs bit string $a$ such that the parity of $a$ equals $f(x)$ with probability $p(a|x)$.
• Figure 2: A non-adaptive distillation protocol for two players using $m$ bipartite NLBs. The PARITY protocol fixes $g$ and $h$ to be the parity function. All boxes receive the same input as the two players.
• Figure 3: The non-adaptive OR protocol for two NLBs.
• Figure 4: The values for various two copy distillation protocols for $\alpha=\frac{1}{2}$. $V_{\textsf{OR}}$ is the highest distilled value for $\epsilon \in \left[ 0,\frac{1}{3}\right]$. Note that even though the curve for $V_{\textsf{OR}}$ remains at the top for $\epsilon < 0$, these values do not correspond to a valid NLB.
• Figure 5: The bounded triangle corresponds to the region where OR is optimal among the known protocols for distilling two copies of correlatedNLBs.

Theorems & Definitions (5)

• Theorem 1
• Theorem 2
• Lemma 1
• Lemma 2
• Theorem 3