Mutually-orthogonal unitary and orthogonal matrices

TL;DR

This paper introduces n-OU and n-OO matrix sets, characterizes order-three n-OO sets, explores their applications in quantum information, and proposes new matrix decomposition methods based on these sets.

Contribution

It provides a detailed characterization of order-three n-OO matrix sets and introduces new matrix decomposition techniques using n-OU and n-OO sets.

Findings

Minimum of three unextendible maximally entangled bases in real two-qutrit systems.

Any order-d matrix admits a d-OU decomposition.

Criteria established for order-three matrices to have n-OO decompositions.

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.