On orthogonal bases in the Hilbert-Schmidt space of matrices

TL;DR

This paper reviews properties and identities of orthogonal matrix bases in finite-dimensional Hilbert-Schmidt spaces, highlighting their importance in quantum physics and information theory.

Contribution

It compiles various facts and identities related to orthogonal bases of matrices in finite-dimensional spaces, emphasizing their applications in quantum information.

Findings

Collection of identities for orthogonal matrix bases

Relevance to quantum states and error correction

Insights into the structure of matrix decompositions

Abstract

Decomposition of (finite-dimensional) operators in terms of orthogonal bases of matrices has been a standard method in quantum physics for decades. In recent years, it has become increasingly popular because of various methodologies applied in quantum information, such as the graph state formalism and the theory of quantum error correcting codes, but also due to the intensified research on the Bloch representation of quantum states. In this contribution we collect various interesting facts and identities that hold for finite-dimensional orthogonal matrix bases.