Local Equivalence of Multipartite Entanglement

Qiao, Y; Sun, X; Yu, N

Local Equivalence of Multipartite Entanglement

Qiao, Y

Sun, X Yu, N

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Publisher:: IEEE-INST ELECTRICAL ELECTRONICS ENGINEERS INC
Publication Type:: Journal Article
Citation:: IEEE Journal on Selected Areas in Communications, 2020, 38, (3), pp. 568-574
Issue Date:: 2020-03-01

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Field	Value	Language
dc.contributor.author	Qiao, Y https://orcid.org/0000-0003-4334-1449
dc.contributor.author	Sun, X
dc.contributor.author	Yu, N https://orcid.org/0000-0003-1188-3032
dc.date.accessioned	2021-02-26T20:49:05Z
dc.date.available	2021-02-26T20:49:05Z
dc.date.issued	2020-03-01
dc.identifier.citation	IEEE Journal on Selected Areas in Communications, 2020, 38, (3), pp. 568-574
dc.identifier.issn	0733-8716
dc.identifier.issn	1558-0008
dc.identifier.uri	http://hdl.handle.net/10453/146499
dc.description.abstract	© 2020 IEEE. Let R be an invariant polynomial ring of a reductive group acting on a vector space, and let d be the minimum integer such that R is generated by those polynomials in R of degree no more than d. To upper bound such d is a long standing open problem since the very initial study of the invariant theory in the 19th century. Motivated by its significant role in characterizing multipartite entanglement, we study the invariant polynomial rings of local unitary groups-the direct product of unitary groups acting on the tensor product of Hilbert spaces, and local general linear groups-the direct product of general linear groups acting on the tensor product of Hilbert spaces. For these two group actions, we prove explicit upper bounds on the degrees needed to generate the corresponding invariant polynomial rings. On the other hand, systematic methods are provided to construct all homogeneous polynomials that are invariant under these two groups for any fixed degree. Thus, our results can be regarded as a complete characterization of the invariant polynomial rings. As an interesting application, we show that multipartite entanglement is additive in the sense that two multipartite states are local unitary equivalent if and only if r-copies of them are local unitary equivalent for some r.
dc.language	English
dc.publisher	IEEE-INST ELECTRICAL ELECTRONICS ENGINEERS INC
dc.relation	http://purl.org/au-research/grants/arc/DE150100720
dc.relation.ispartof	IEEE Journal on Selected Areas in Communications
dc.relation.isbasedon	10.1109/JSAC.2020.2969004
dc.rights	info:eu-repo/semantics/closedAccess
dc.rights	© 2020 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other uses, in any current or future media, including reprinting/republishing this material for advertising or promotional purposes, creating new collective works, for resale or redistribution to servers or lists, or reuse of any copyrighted component of this work in other works.	en_US
dc.subject	0805 Distributed Computing, 0906 Electrical and Electronic Engineering, 1005 Communications Technologies
dc.subject.classification	Networking & Telecommunications
dc.title	Local Equivalence of Multipartite Entanglement
dc.type	Journal Article
utslib.citation.volume	38
utslib.for	0805 Distributed Computing
utslib.for	0906 Electrical and Electronic Engineering
utslib.for	1005 Communications Technologies
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
pubs.organisational-group	/University of Technology Sydney
pubs.organisational-group	/University of Technology Sydney/Faculty of Engineering and Information Technology/School of Computer Science
utslib.copyright.status	closed_access	*
dc.date.updated	2021-02-26T20:49:03Z
pubs.issue	3
pubs.publication-status	Published
pubs.volume	38
utslib.citation.issue	3

Abstract:

© 2020 IEEE. Let R be an invariant polynomial ring of a reductive group acting on a vector space, and let d be the minimum integer such that R is generated by those polynomials in R of degree no more than d. To upper bound such d is a long standing open problem since the very initial study of the invariant theory in the 19th century. Motivated by its significant role in characterizing multipartite entanglement, we study the invariant polynomial rings of local unitary groups-the direct product of unitary groups acting on the tensor product of Hilbert spaces, and local general linear groups-the direct product of general linear groups acting on the tensor product of Hilbert spaces. For these two group actions, we prove explicit upper bounds on the degrees needed to generate the corresponding invariant polynomial rings. On the other hand, systematic methods are provided to construct all homogeneous polynomials that are invariant under these two groups for any fixed degree. Thus, our results can be regarded as a complete characterization of the invariant polynomial rings. As an interesting application, we show that multipartite entanglement is additive in the sense that two multipartite states are local unitary equivalent if and only if r-copies of them are local unitary equivalent for some r.

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http://hdl.handle.net/10453/146499