[论文解读] Multiplayer XOR games and quantum communication complexity with clique-wise entanglement

Jop Briët, Harry Buhrman+2 more|ArXiv.org|Nov 20, 2009

Complexity and Algorithms in Graphs参考文献 34被引用 20

一句话总结

该论文证明，在多人XOR游戏中，当玩家共享Schmidt态或团状纠缠（GHZ与EPR态的组合）时，量子优势——以纠缠与经典偏差之比衡量——被一个常数所限制。作者利用多线性Grothendieck不等式及稳定子态的结构性结果，证明此类纠缠模式无法产生无界的量子优势，从而解决了算子代数领域一个35年来的未解问题，并将差异性方法扩展至含纠缠的多体量子通信复杂性。

ABSTRACT

XOR games are a simple computational model with connections to many areas of complexity theory. Perhaps the earliest use of XOR games was in the study of quantum correlations. XOR games also have an interesting connection to Grothendieck's inequality, a fundamental theorem of analysis, which shows that two players sharing entanglement can achieve at most a constant factor advantage over players following classical strategies in an XOR game. Perez-Garcia et al. show that when the players share GHZ states, this advantage is bounded by a constant. We use a multilinear generalization of Grothendieck's inequality due to Blei and Tonge to simplify the proof of the second result and extend it to the case of so-called Schmidt states, answering an open problem of Perez-Garcia et al. Via a reduction given in that paper, this answers a 35-year-old problem in operator algebras due to Varopoulos, showing that the space of compact operators on a Hilbert space is a Q-algebra under Schur product. A further generalization of Grothendieck's inequality due to Carne lets us show that the gap between the entangled and classical value is at most a constant in any multiplayer XOR game in which the players are allowed to share combinations of GHZ states and EPR pairs of any dimension. As an application of our results, we show that the discrepancy method in communication complexity remains a lower bound in the multiparty model where the players have quantum communication and the kinds of entanglement discussed above. This answers an open question of Lee, Schechtman, and Shraibman.

研究动机与目标

• 理解在特定纠缠结构下，多人XOR游戏中量子优势的极限。
• 解决Pérez-García等人提出的关于Schmidt态下量子-经典偏差差距有界的开放性问题。
• 将Grothendieck型不等式的适用范围扩展至多体情形，并将其应用于量子通信复杂性。
• 证明三体系统中的稳定子态仅能带来常数因子的量子优势。
• 确立差异性方法在含GHZ、EPR及稳定子态的多体量子通信复杂性中仍为有效下界。

提出的方法

• 利用Blei与Tonge提出的Grothendieck不等式的多线性推广，来界定共享Schmidt态的XOR游戏中偏差。
• 应用Carne对Grothendieck不等式的推广，分析混合GHZ与EPR纠缠的博弈。
• 将稳定子态的分析简化为图态，并通过酉变换将其转化为团状纠缠态。
• 采用张量分解方法，将偏差表达为纠缠态上广义内积的形式。
• 利用图态及其与GHZ/EPR类态的酉等价结构，界定可实现偏差的最大值。
• 通过从博弈论设定的约化，应用算子代数中的结果，特别是Varopoulos的Q代数猜想。

实验结果

研究问题

• RQ1 在多人XOR游戏中，能否通过Schmidt态（GHZ态的推广）实现无界的量子优势？
• RQ2 当玩家仅共享GHZ态或GHZ与EPR对的组合时，量子-经典偏差差距是否仍被限制？
• RQ3 在允许纠缠的情况下，差异性方法在多体量子通信复杂性中是否仍为有效下界？
• RQ4 能否利用稳定子态的结构来界定三体XOR游戏中量子优势的上界？
• RQ5 根据结果，希尔伯特空间上紧算子空间在Schur积下是否具有Q代数结构？

主要发现

• 当玩家共享Schmidt态时，任何多人XOR游戏中量子-经典偏差差距均被一个常数所限制，从而解决了Pérez-García等人提出的开放性问题。
• 对于任意子集内GHZ与EPR对的组合，该有界性依然成立，意味着即使存在任意团状纠缠，量子优势也仅为常数因子。
• 当玩家共享GHZ态、EPR对或稳定子态时，差异性方法在多体量子通信复杂性中仍为有效下界。
• 结果表明，三体稳定子态在XOR游戏中无法提供超过常数因子的量子优势，相较于无纠缠策略。
• 本文通过从博弈论设定的约化，解决了算子代数领域一个35年未解的问题，证明了紧算子空间在Schur积下为Q代数。