LOCC protocols with bounded width per round optimize convex functions

Author:

Leung Debbie12ORCID,Winter Andreas3ORCID,Yu Nengkun14ORCID

Affiliation:

1. Institute for Quantum Computing and Department of Combinatorics and Optimization, University of Waterloo, Waterloo, Ontario, Canada

2. Perimeter Institute for Theoretical Physics, Waterloo, Ontario, Canada

3. ICREA & Física Teórica: Informació i Fenómens Quàntics, Departament de Física, Universitat Autònoma de Barcelona, ES-08193 Bellaterra, Barcelona, Spain

4. Centre for Quantum Software and Information, Faculty of Engineering and Information Technology, University of Technology Sydney, NSW 2007, Australia

Abstract

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 [Formula: see text]-dimensional local system, is replaced by one with at most [Formula: see text] 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 [Formula: see text]-dimensional local system has at most [Formula: see text] 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 [Formula: see text] coarse-grained final measurement outcomes, global input dimension [Formula: see text] and global output dimension [Formula: see text] for [Formula: see text] conditioned on the [Formula: see text]th outcome, then one can obtain the instrument as a convex combination of no more than [Formula: see text] slim protocols; that is, [Formula: see text] bits of shared randomness suffice.

Funder

FEDER

CIFAR

Natural Sciences and Engineering Research Council of Canada

Spanish MINECO

Generalitat de Catalunya

ARC

Publisher

World Scientific Pub Co Pte Lt

Subject

Mathematical Physics,Statistical and Nonlinear Physics

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