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Matrix Product Representation of Locality Preserving Unitaries

Şahinoglu, M. Burak and Shukla, Sujeet K. and Bi, Feng and Chen, Xie (2018) Matrix Product Representation of Locality Preserving Unitaries. Physical Review B, 98 (24). Art. No. 245122. ISSN 2469-9950. http://resolver.caltech.edu/CaltechAUTHORS:20171108-141516563

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Abstract

Matrix product representation provides a useful formalism to study not only entangled states but also entangled operators in one dimension. In this paper, we focus on unitary transformations and show that matrix product operators that are unitary provide a necessary and sufficient representation of one-dimensional (1D) unitaries that preserve locality. That is, we show that matrix product operators that are unitary are guaranteed to preserve locality by mapping local operators to local operators, while at the same time all locality-preserving unitaries can be represented in a matrix product way. Moreover, we show that matrix product representation gives a straightforward way to extract the index defined by Gross, Nesme, Vogts, and Werner in [D. Gross et al., Commun. Math. Phys. 310, 419 (2012)] for classifying 1D locality-preserving unitaries. The key to our discussion is a set of “fixed-point” conditions which characterize the form of the matrix product unitary operators after blocking sites. Finally, we show that if the unitary condition is only required for certain system sizes, then matrix product formalism allows more possibilities. In particular, we give an example of a simple matrix product operator which is unitary only for odd system sizes, does not preserve locality, and carries a “fractional” index.


Item Type:Article
Related URLs:
URLURL TypeDescription
https://doi.org/10.1103/PhysRevB.98.245122DOIArticle
https://arxiv.org/abs/1704.01943arXivDiscussion Paper
ORCID:
AuthorORCID
Shukla, Sujeet K.0000-0002-4283-536X
Additional Information:© 2018 American Physical Society. Received 17 May 2018; published 17 December 2018. While working on this project, we became aware of similar work being carried out by J. I. Cirac, D. Perez-Garcia, N. Schuch, and F. Verstraete which has been reported in Ref. [27]. We acknowledge helpful discussions with Lukasz Fidkowski that inspired this project. X.C. is supported by the National Science Foundation under Award No. DMR-1654340. We acknowledge funding provided by the Walter Burke Institute for Theoretical Physics and the Institute for Quantum Information and Matter, an NSF Physics Frontiers Center (NSF Grant No. PHY-1125565) with support of the Gordon and Betty Moore Foundation (Grant No. GBMF-2644). M.B.S. is supported by the Simons Foundation through the It from Qubit collaboration.
Group:IQIM, Institute for Quantum Information and Matter, Walter Burke Institute for Theoretical Physics
Funders:
Funding AgencyGrant Number
NSFDMR-1654340
Walter Burke Institute for Theoretical Physics, CaltechUNSPECIFIED
Institute for Quantum Information and Matter (IQIM)UNSPECIFIED
NSFPHY-1125565
Gordon and Betty Moore FoundationGBMF-2644
Simons FoundationUNSPECIFIED
Record Number:CaltechAUTHORS:20171108-141516563
Persistent URL:http://resolver.caltech.edu/CaltechAUTHORS:20171108-141516563
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:83074
Collection:CaltechAUTHORS
Deposited By: Bonnie Leung
Deposited On:08 Nov 2017 22:52
Last Modified:23 Jan 2019 22:16

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