[论文解读] No Strong Parallel Repetition with Entangled and Non-signaling Provers

Julia Kempe, Oded Regev|ArXiv.org|Nov 1, 2009

Complexity and Algorithms in Graphs参考文献 16被引用 4

一句话总结

本文证明了当两个证明者共享纠缠或为非信号时，强并行重复不成立，推翻了一个长期存在的猜想。通过使用唯一游戏和线性游戏构造显式反例，表明获胜概率衰减为 (1−Θ(1/n²))ℓ，证明了对非信号证明者的 Hölzinger 边界以及对纠缠证明者的已知边界是紧致的。

ABSTRACT

We consider one-round games between a classical verifier and two provers. One of the main questions in this area is the \emph{parallel repetition question}: If the game is played $\ell$ times in parallel, does the maximum winning probability decay exponentially in $\ell$? In the classical setting, this question was answered in the affirmative by Raz. More recently the question arose whether the decay is of the form $(1-Θ(\eps))^\ell$ where $1-\eps$ is the value of the game and $\ell$ is the number of repetitions. This question is known as the \emph{strong parallel repetition question} and was motivated by its connections to the unique games conjecture. It was resolved by Raz who showed that strong parallel repetition does \emph{not} hold, even in the very special case of games known as XOR games. This opens the question whether strong parallel repetition holds in the case when the provers share entanglement. Evidence for this is provided by the behavior of XOR games, which have strong (in fact \emph{perfect}) parallel repetition, and by the recently proved strong parallel repetition of linear unique games. A similar question was open for games with so-called non-signaling provers. Here the best known parallel repetition theorem is due to Holenstein, and is of the form $(1-Θ(\eps^2))^\ell$. We show that strong parallel repetition holds neither with entangled provers nor with non-signaling provers. In particular we obtain that Holenstein's bound is tight. Along the way we also provide a tight characterization of the asymptotic behavior of the entangled value under parallel repetition of unique games in terms of a semidefinite program.

研究动机与目标

• 解决纠缠和非信号证明者游戏中强并行重复的问题。
• 确定在纠缠或非信号证明者参与的游戏中，重复游戏的获胜概率是否以 (1−ε)ℓ 的形式指数衰减。
• 在并行重复下，建立纠缠值和非信号值衰减速率的紧致边界。
• 通过半定规划表征唯一游戏中纠缠值的渐近行为。
• 通过构造紧致反例，证明非信号和纠缠证明者现有的并行重复定理是最优的。

提出的方法

• 构造一族唯一游戏，即“唯一线性游戏” G_uL，其经典值 ω(G_uL) = 1−Θ(1/n)，纠缠值 ω*(G_uL) = 1−Θ(1/n²)，非信号值 ω^ns(G_uL) = 1−Θ(1/n²)。
• 使用半定规划（SDP1 和 SDP2）表征唯一游戏中纠缠值，表明 SDP1 和 SDP2 解之间存在二次差距。
• 基于相关采样和从 SDP 向量导出的正交基，应用张量积策略，构建重复游戏的获胜策略。
• 通过证明该策略在 G_L^ℓ 中从不使用答案 3 且保持获胜概率，表明其在 G_L^ℓ 中的有效性。
• 利用 G_uL 的 SDP 解中内积非负的性质，确保其在 SDP2 约束下的可行性。
• 通过证明当 ℓ ≥ n² 时，ω^ns(G_L^ℓ) ≥ (1−O(1/n²))ℓ，证明 Hölzinger 边界的紧致性，与非信号值的衰减速率一致。

实验结果

研究问题

• RQ1 当证明者共享纠缠时，强并行重复是否对两证明者游戏成立？
• RQ2 在非信号值的并行重复下，衰减速率是否为 (1−Θ(ε))ℓ 的形式，还是更慢？
• RQ3 Hölzinger 对非信号证明者的并行重复边界是否可改进，或其本身已是紧致的？
• RQ4 在并行重复下，唯一游戏中纠缠值的渐近行为如何？
• RQ5 XOR 游戏或其他特殊类别的游戏在纠缠或非信号设置下是否表现出完美或强并行重复？

主要发现

• 通过构造一族唯一游戏，证明纠缠证明者的强并行重复不成立，其纠缠值衰减为 (1−Θ(1/n²))ℓ。
• 同一游戏的非信号值也衰减为 (1−Θ(1/n²))ℓ，证明 Hölzinger 的边界 (1−Θ(ε²))ℓ 是紧致的。
• 线性游戏 G_L 为强并行重复提供了一个字母表大小为 2 的反例，表明即使简单游戏也会违反强并行重复。
• 唯一游戏中 SDP1 与 SDP2 松弛之间存在二次差距，表明在此背景下 SDP2 严格强于 SDP1。
• 所构造的重复游戏策略并非乘积策略，尽管其基于 SDP 解的张量积，但因包含相关采样步骤而具有相关性。
• 当 ℓ ≥ n² 时，G_L^ℓ 的经典值、纠缠值和非信号值均满足 ω(G_L^ℓ) ≥ (1−O(1/n²))ℓ，表明该衰减速率是游戏结构的固有属性。