We start with the task of discriminating finitely many multipartite quantum states using LOCC protocols, with the goal to optimize the probability of correctly identifying the state. We provide two different methods to show that finitely many measurement outcomes in every step are sufficient for approaching the optimal probability of discrimination. In the first method, each measurement of an optimal LOCC protocol, applied to a dloc-dimensional local system, is replaced by one with at most 2dloc2 outcomes, without changing the probability of success. In the second method, we decompose any LOCC protocol into a convex combination of a number of “slim protocols” in which each measurement applied to a dloc-dimensional local system has at most dloc2 outcomes. To maximize any convex functions in LOCC (including the probability of state discrimination or fidelity of state transformation), an optimal protocol can be replaced by the best slim protocol in the convex decomposition without using shared randomness. For either method, the bound on the number of outcomes per measurement is independent of the global dimension, the number of parties, the depth of the protocol, how deep the measurement is located, and applies to LOCC protocols with infinite rounds, and the “measurement compression” can be done “top-down” — independent of later operations in the LOCC protocol. The second method can be generalized to implement LOCC instruments with finitely many classical outcomes: if the instrument has n coarse-grained final measurement outcomes, global input dimension D0 and global output dimension Di for i = 1,…,n conditioned on the ith outcome, then one can obtain the instrument as a convex combination of no more than R = 1 − D02 +∑ i=1nD 02D i2 slim protocols; that is, log2R bits of shared randomness suffice.

中文翻译：

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每轮有界宽度的 LOCC 协议优化凸函数

我们从使用 LOCC 协议识别有限多个多方量子状态的任务开始，目标是优化正确识别状态的概率。我们提供了两种不同的方法来证明每个步骤中有限的多个测量结果足以接近最佳的辨别概率。在第一种方法中，最优 LOCC 协议的每次测量，应用于d位置维局部系统，最多被替换为一个2d位置2结果，而不改变成功的概率。在第二种方法中，我们分解任何LOCC 协议为多个“精简协议”的凸组合，其中每个测量应用于一个d位置维局部系统至多有d位置2结果。为了最大化 LOCC 中的任何凸函数（包括状态区分的概率或状态转换的保真度），可以在不使用共享随机性的情况下将最优协议替换为凸分解中的最佳苗条协议。对于任何一种方法，每次测量的结果数量的界限都与全局维度、参与方的数量、协议的深度、测量的深度无关，并且适用于无限轮次的 LOCC 协议，以及“测量压缩”可以“自上而下”完成——独立于 LOCC 协议中的后续操作。第二种方法可以推广到实现具有有限多个经典结果的 LOCC 工具：如果工具具有n粗粒度最终测量结果，全局输入维度D0和全局输出维度D一世为了一世 = 1,…,n条件为一世th 结果，则可以将工具作为不超过的凸组合R = 1 - D02 +∑ 一世=1nD 02D 一世2精简协议；那是，日志2R一些共享随机性就足够了。