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arXiv · 2309.11128

Mutually-orthogonal unitary and orthogonal matrices

Zhiwei Song·Lin Chen·Saiqi Liu
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Abstract

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.

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Zhiwei Song, Lin Chen, Saiqi Liu. 2023-09-20. Mutually-orthogonal unitary and orthogonal matrices. https://doi.org/10.1016/j.laa.2025.06.025

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