[论文解读] Decidability of Fully Quantum Nonlocal Games with Noisy Maximally Entangled States

Minglong Qin, Penghui Yao|arXiv (Cornell University)|Nov 19, 2022

Quantum Mechanics and Applications被引用 3

一句话总结

本文证明了当玩家共享噪声最大纠缠态时，完全量子非局部游戏的可判定性，证明了为任意精确逼近量子值所需噪声EPR对数目的可计算上界。通过将傅里叶分析推广至超算子，并证明不变性原理与维数约化，作者表明共享纠缠中的噪声恢复了可计算性，与无噪声情形的不可判定性（MIP*=RE）形成对比。

ABSTRACT

This paper considers the decidability of fully quantum nonlocal games with noisy maximally entangled states. Fully quantum nonlocal games are a generalization of nonlocal games, where both questions and answers are quantum and the referee performs a binary POVM measurement to decide whether they win the game after receiving the quantum answers from the players. The quantum value of a fully quantum nonlocal game is the supremum of the probability that they win the game, where the supremum is taken over all the possible entangled states shared between the players and all the valid quantum operations performed by the players. The seminal work $\mathrm{MIP}^*=\mathrm{RE}$ implies that it is undecidable to approximate the quantum value of a fully nonlocal game. This still holds even if the players are only allowed to share (arbitrarily many copies of) maximally entangled states. This paper investigates the case that the shared maximally entangled states are noisy. We prove that there is a computable upper bound on the copies of noisy maximally entangled states for the players to win a fully quantum nonlocal game with a probability arbitrarily close to the quantum value. This implies that it is decidable to approximate the quantum values of these games. Hence, the hardness of approximating the quantum value of a fully quantum nonlocal game is not robust against the noise in the shared states. This paper is built on the framework for the decidability of non-interactive simulations of joint distributions and generalizes the analogous result for nonlocal games. We extend the theory of Fourier analysis to the space of super-operators and prove several key results including an invariance principle and a dimension reduction for super-operators. These results are interesting in their own right and are believed to have further applications.

研究动机与目标

• 研究当共享纠缠为噪声态时，完全量子非局部游戏中量子值逼近的不可判定性是否仍然存在。
• 确定噪声最大纠缠态是否允许对逼近量子值所需的副本数建立可计算上界。
• 将非交互式模拟与傅里叶分析的框架推广至量子非局部游戏背景下的超算子空间。
• 确立共享纠缠中的噪声使量子值逼近的复杂性降低，恢复可判定性。

提出的方法

• 构建超算子傅里叶分析框架，以分析完全量子非局部游戏中量子策略的性质。
• 证明超算子的不变性原理与维数约化定理，将经典不变性原理推广至量子算子空间。
• 利用Choi-Jamiołkowski同构将量子操作表示为算子，并应用迹范数与谱分解。
• 应用Hölder不等式与迹不等式，以界定投影前后超算子之间的差异。
• 引入噪声算子∆γ(P)以模拟最大纠缠态中的退相干，并分析其对策略性能的影响。
• 采用ε-网与算子范数界，证明仅需有限数量的噪声EPR对即可在任意ε内逼近量子值。

实验结果

研究问题

• RQ1 当玩家仅共享噪声最大纠缠态而非理想EPR对时，完全量子非局部游戏的量子值是否可判定？
• RQ2 能否对实现量子值ε-逼近所需噪声EPR对数建立可计算上界？
• RQ3 与无噪声情形相比，共享纠缠中的噪声如何影响逼近量子非局部游戏值的复杂性？
• RQ4 经典工具如傅里叶分析在多大程度上可推广至量子信息中的超算子空间？
• RQ5 超算子的不变性原理与维数约化在噪声量子操作下是否依然成立？

主要发现

• 当玩家共享噪声最大纠缠态时，完全量子非局部游戏的量子值是可判定的，因为存在一个可计算的上界，使得在任意ε > 0内逼近该值所需的噪声EPR对数有限。
• 作者证明了超算子不变性原理与维数约化结果，这些是实现可判定性的关键技术工具。
• 共享纠缠中的噪声恢复了可计算性：与MIP*=RE结果（理想EPR对导致不可判定性）相反，噪声态使问题变为可判定。
• 本文确立了可通过有限维策略空间上的ε-网逼近量子值，其界由噪声EPR对的数量决定。
• 对噪声EPR对数的上界仅依赖于游戏的描述与所需精度ε，而不依赖于底层态空间的复杂性。
• 结果推广了先前关于具有噪声纠缠的经典非局部游戏的研究，并将非交互式模拟理论扩展至完全量子情形。