LOCC protocols with bounded width per round optimize convex functions

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.