Abstract
In this work we consider the role of entanglement assistance in quantum communication protocols, focusing, in particular, on whether the type of shared entangled state can affect the quantum communication complexity of a function. This question is interesting because in some other settings in quantum information, such as nonlocal games, or tasks that involve quantum communication between players and referee, or simulating bipartite unitaries or communication channels, maximally entangled states are known to be less useful as a resource than some partially entangled states. By contrast, we prove that the boundederror entanglementassisted quantum communication complexity of a partial or total function cannot be improved by more than a constant factor by replacing maximally entangled states with arbitrary entangled states. In particular, we show that every quantum communication protocol using Q qubits of communication and arbitrary shared entanglement can be epsilonapproximated by a protocol using O(Q/epsilon+log(1/epsilon)/epsilon) qubits of communication and only EPR pairs as shared entanglement. This conclusion is opposite of the common wisdom in the study of nonlocal games, where it has been shown, for example, that the I3322 inequality has a nonlocal strategy using a nonmaximally entangled state, which surpasses the winning probability achievable by any strategy using a maximally entangled state of any dimension [Vidick and Wehner, 2011]. We leave open the question of how much the use of a shared maximally entangled state can reduce the quantum communication complexity of a function.
Our second result concerns an old question in quantum information theory: How much quantum communication is required to approximately convert one pure bipartite entangled state into another? We give simple and efficiently computable upper and lower bounds. Given two bipartite states chi> and upsilon>, we define a natural quantity, d_{infty}(chi>, upsilon>), which we call the l_{infty} Earth Mover's distance, and we show that the communication cost of converting between chi> and upsilon> is upper bounded by a constant multiple of d_{infty}(chi>, upsilon>). Here d_{infty}(chi>, upsilon>) may be informally described as the minimum over all transports between the log of the Schmidt coefficients of chi> and those of upsilon>, of the maximum distance that any amount of mass must be moved in that transport. A precise definition is given in the introduction. Furthermore, we prove a complementary lower bound on the cost of state conversion by the epsilonSmoothed l_{infty}Earth Mover's Distance, which is a natural smoothing of the l_{infty}Earth Mover's Distance that we will define via a connection with optimal transport theory.
BibTeX  Entry
@InProceedings{coudron_et_al:LIPIcs:2019:10842,
author = {Matthew Coudron and Aram W. Harrow},
title = {{Universality of EPR Pairs in EntanglementAssisted Communication Complexity, and the Communication Cost of State Conversion}},
booktitle = {34th Computational Complexity Conference (CCC 2019)},
pages = {20:120:25},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959771160},
ISSN = {18688969},
year = {2019},
volume = {137},
editor = {Amir Shpilka},
publisher = {Schloss DagstuhlLeibnizZentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2019/10842},
URN = {urn:nbn:de:0030drops108421},
doi = {10.4230/LIPIcs.CCC.2019.20},
annote = {Keywords: Entanglement, quantum communication complexity}
}
Keywords: 

Entanglement, quantum communication complexity 
Seminar: 

34th Computational Complexity Conference (CCC 2019) 
Issue Date: 

2019 
Date of publication: 

16.07.2019 