LOCC protocols with bounded width per round optimize convex functions

Leung, D; Winter, A; Yu, N

LOCC protocols with bounded width per round optimize convex functions

Leung, D Winter, A Yu, N

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Publisher:: World Scientific Publishing
Publication Type:: Journal Article
Citation:: Reviews in Mathematical Physics: a journal for survey and expository articles in the field of mathematical physics, 2021, 33, (5)
Issue Date:: 2021-01-30

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Field	Value	Language
dc.contributor.author	Leung, D
dc.contributor.author	Winter, A
dc.contributor.author	Yu, N https://orcid.org/0000-0003-1188-3032
dc.date.accessioned	2022-03-18T04:59:53Z
dc.date.available	2022-03-18T04:59:53Z
dc.date.issued	2021-01-30
dc.identifier.citation	Reviews in Mathematical Physics: a journal for survey and expository articles in the field of mathematical physics, 2021, 33, (5)
dc.identifier.issn	0129-055X
dc.identifier.issn	1793-6659
dc.identifier.uri	http://hdl.handle.net/10453/155344
dc.description.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 dloc-dimensional local system, is replaced by one with at most 2d2loc 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 d2loc 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−D20+∑ni=1D20D2i slim protocols; that is, log2R bits of shared randomness suffice.
dc.language	English
dc.publisher	World Scientific Publishing
dc.relation	http://purl.org/au-research/grants/arc/DE180100156
dc.relation	http://purl.org/au-research/grants/arc/DP210102449
dc.relation.ispartof	Reviews in Mathematical Physics: a journal for survey and expository articles in the field of mathematical physics
dc.relation.isbasedon	10.1142/s0129055x21500136
dc.rights	info:eu-repo/semantics/openAccess
dc.subject	01 Mathematical Sciences, 02 Physical Sciences
dc.subject.classification	Mathematical Physics
dc.title	LOCC protocols with bounded width per round optimize convex functions
dc.type	Journal Article
utslib.citation.volume	33
utslib.for	01 Mathematical Sciences
utslib.for	02 Physical Sciences
pubs.organisational-group	/University of Technology Sydney
pubs.organisational-group	/University of Technology Sydney/Faculty of Engineering and Information Technology
pubs.organisational-group	/University of Technology Sydney/Strength - QSI - Centre for Quantum Software and Information
utslib.copyright.status	open_access	*
pubs.consider-herdc	false
dc.date.updated	2022-03-18T04:59:53Z
pubs.issue	5
pubs.publication-status	Published
pubs.volume	33
utslib.citation.issue	5

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 dloc-dimensional local system, is replaced by one with at most 2d2loc 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 d2loc 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−D20+∑ni=1D20D2i slim protocols; that is, log2R bits of shared randomness suffice.

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