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arXiv 1410.4397quant-ph
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自由纠缠博弈的并行重复：简化与改进

Parallel Repetition of Free Entangled Games: Simplification and Improvements

André Chailloux, Giannicola Scarpa

备注：
17 pages, this paper is a follow up and supersedes our previous paper 'Parallel Repetition of Entangled Games with Exponential Decay via the Superposed Information Cost' [CS14, arXiv:1310.7787] v2: updated GS affiliation
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摘要：

在两人博弈中，两个合作但不交流的参与者，爱丽丝和鲍勃，接收来自概率分布的输入。他们每个人都产生一个输出，如果他们的输入/输出满足某个谓词，他们就赢得了比赛。一个游戏$G$的纠缠值$ω^*(G)$是Alice和Bob在接收到输入之前共享纠缠态时赢得游戏的最大概率。$G$的$n$倍并行重复$G^n$由$G$的$n$实例组成，其中Alice和Bob同时接收所有输入并且必须同时产生所有输出。如果它们赢得了$G$的每个实例，则它们将赢得$G^n$。最近，有一系列的作品显示了投影游戏[DSV 13]，均匀分布游戏[CS 14]和免费游戏（即产品分布游戏[JPY 13]）的指数衰减平行重复。这篇文章是[CS 14]的后续，我们改进和简化了前一篇文章的几个部分。我们的主要结果是，对于任何值为$ω^*(G)=1-\varepsilon$的免费游戏$G$，我们都有$ω^*(G^n) \le (1 - \varepsilon^2)^{Ω(\frac{n}{\log(l)})}$，其中$l$是游戏输出集的大小。该结果改进了[JPY 13]和[CS 14]中的结果。我们使用的框架也可以扩展到免费投影游戏。我们表明，对于一个免费的投影游戏$G$值$ω^*(G)=1-\varepsilon$，我们有$ω^*(G^n) \le (1 - \varepsilon)^{Ω(n)}$。

英文摘要：

In a two-player game, two cooperating but non communicating players, Alice and Bob, receive inputs taken from a probability distribution. Each of them produces an output and they win the game if they satisfy some predicate on their inputs/outputs. The entangled value $ω^*(G)$ of a game $G$ is the maximum probability that Alice and Bob can win the game if they are allowed to share an entangled state prior to receiving their inputs. The $n$-fold parallel repetition $G^n$ of $G$ consists of $n$ instances of $G$ where Alice and Bob receive all the inputs at the same time and must produce all the outputs at the same time. They win $G^n$ if they win each instance of $G$. Recently, there has been a series of works showing parallel repetition with exponential decay for projection games [DSV13], games on the uniform distribution [CS14] and for free games, i.e. games on a product distribution [JPY13]. This article is meant to be a follow up of [CS14], where we improve and simplify several parts of our previous paper. Our main result is that for any free game $G$ with value $ω^*(G)=1-\varepsilon$, we have $ω^*(G^n) \le (1 - \varepsilon^2)^{Ω(\frac{n}{\log(l)})}$ where $l$ is the size of the output set of the game. This result improves on both the results in [JPY13] and [CS14]. The framework we use can also be extended to free projection games. We show that for a free projection game $G$ with value $ω^*(G)=1-\varepsilon$, we have $ω^*(G^n) \le (1 - \varepsilon)^{Ω(n)}$.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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