论文标题
具有最大纠缠状态的全量子非本地游戏的可判决性
Decidability of fully quantum nonlocal games with noisy maximally entangled states
论文作者
论文摘要
本文考虑了具有嘈杂的最大纠缠状态的完全量子非本地游戏的可决定性。完全量子的非本地游戏是对非本地游戏的概括，在该游戏中，问题和答案都是量子，裁判进行了二进制POVM测量，以决定他们在从玩家那里获得量子答案后是否赢得了游戏。完全量子非本地游戏的量子价值是他们赢得游戏的可能性的至高无上的，即在玩家之间共享的所有可能的纠缠状态以及玩家执行的所有有效的量子操作之间。开创性的工作$ \ mathrm {mip}^*= \ mathrm {re} $意味着不可证明是无法近似完全非局部游戏的量子值。即使只允许玩家共享（任意多个副本）最大纠缠的状态，这仍然存在。本文调查了共享最大纠结状态的案例是嘈杂的。我们证明，嘈杂的最大纠缠状态的副本上有一个可计算的上限，以供玩家赢得一个完全量子非局部游戏，概率是任意接近量子值的概率。这意味着可以决定这些游戏的量子值。因此，近似完全量子非局部游戏的量子值的硬度与共享状态中的噪声并不强大。 本文建立在框架上，用于对联合分布的非相互作用模拟的确定性，并概括了非本地游戏的类似结果。我们将傅立叶分析的理论扩展到超级操作员的空间，并证明了几个关键结果，包括不变性原理和降低超级操作员的尺寸。这些结果本身很有趣，被认为具有进一步的应用。
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.
