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Mutually orthogonal unitary and orthogonal matrices

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Title: Mutually orthogonal unitary and orthogonal matrices
Authors: Zhiwei Song, Lin Chen, Saiqi Liu
Source: Linear Algebra and its Applications. 725:1-17
Publication Status: Preprint
Publisher Information: Elsevier BV, 2025.
Publication Year: 2025
Subject Terms: Quantum Physics, FOS: Physical sciences, Mathematical Physics (math-ph), Quantum Physics (quant-ph), Mathematical Physics
Description: We introduce the concept of n-OU and n-OO matrix sets, a collection of n mutually-orthogonal unitary and real orthogonal matrices under Hilbert-Schmidt inner product. We give a detailed characterization of order-three n-OO matrix sets under orthogonal equivalence. As an application in quantum information theory, we show that the minimum and maximum numbers of an unextendible maximally entangled bases within a real two-qutrit system are three and four, respectively. Further, we propose a new matrix decomposition approach, defining an n-OU (resp. n-OO) decomposition for a matrix as a linear combination of n matrices from an n-OU (resp. n-OO) matrix set. We show that any order-d matrix has a d-OU decomposition. As a contrast, we provide criteria for an order-three real matrix to possess an n-OO decomposition.
16 pages, no figure
Document Type: Article
Language: English
ISSN: 0024-3795
DOI: 10.1016/j.laa.2025.06.025
DOI: 10.48550/arxiv.2309.11128
Access URL: http://arxiv.org/abs/2309.11128
Rights: Elsevier TDM
arXiv Non-Exclusive Distribution
Accession Number: edsair.doi.dedup.....3ff74d3d02adb51f11eb41b8f559948d
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  Data: We introduce the concept of n-OU and n-OO matrix sets, a collection of n mutually-orthogonal unitary and real orthogonal matrices under Hilbert-Schmidt inner product. We give a detailed characterization of order-three n-OO matrix sets under orthogonal equivalence. As an application in quantum information theory, we show that the minimum and maximum numbers of an unextendible maximally entangled bases within a real two-qutrit system are three and four, respectively. Further, we propose a new matrix decomposition approach, defining an n-OU (resp. n-OO) decomposition for a matrix as a linear combination of n matrices from an n-OU (resp. n-OO) matrix set. We show that any order-d matrix has a d-OU decomposition. As a contrast, we provide criteria for an order-three real matrix to possess an n-OO decomposition.<br />16 pages, no figure
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