Mutually-orthogonal unitary and orthogonal matrices

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.

arXiv comment16 pages, no figure

References in corpus (10)

• Classification of multipartite entanglement in all dimensions
• Unextendible maximally entangled bases and mutually unbiased bases
• Two-Unitary Decomposition Algorithm and Open Quantum System Simulation
• Unextendible maximally entangled bases in ${\mathbb{C}}^d⨂{\mathbb{C}}^{d^{\prime }}$
• Unextendible maximally entangled bases in dxd
• On the mixed-unitary rank of quantum channels
• Constructing UMEB from maximally entangled basis
• SLOCC Invariants for Multipartite Mixed States
• Unitary and orthogonal equivalence of sets of matrices
• Simultaneous unitary equivalences