纠缠和无信号证明器的无强并行重复
No Strong Parallel Repetition with Entangled and Non-signaling Provers
摘要：
我们考虑一个经典的验证者和两个证明者之间的单轮博弈。这方面的一个主要问题是\emph{parallel repetition question}：如果游戏并行进行$\ell$次，最大获胜概率是否在$\ell$中呈指数衰减？在经典的背景下，拉兹对这个问题的回答是肯定的。最近出现的问题是否衰减的形式$(1-Θ(\eps))^\ell$其中$1-\eps$是游戏的价值和$\ell$是重复的次数。这个问题被称为\emph{strong parallel repetition question}，其动机是它与独特的游戏猜想的联系。这是解决拉兹谁表明，强平行重复不\emph{not}举行，即使在非常特殊的情况下，游戏称为异或游戏。这就引出了一个问题，即当证明者共享纠缠时，强平行重复是否成立。XOR游戏的行为提供了这方面的证据，XOR游戏具有很强的（实际上是\emph{perfect}）并行重复，并且最近证明了线性独特游戏的强并行重复。一个类似的问题是开放的游戏与所谓的非信号证明。这里最著名的平行重复定理是由Holenstein提出的，其形式为$(1-Θ(\eps^2))^\ell$。我们表明，强并行重复既不与纠缠证明，也不与非信号证明。特别是我们得到Holenstein的界是紧的。一路上，我们还提供了一个紧密的纠缠值的渐近行为下的并行重复独特的游戏在一个半定程序的表征。
英文摘要：
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.
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