Parametrization of degenerate density matrices

TL;DR

This paper introduces a parametrization method for degenerate density matrices using spectral representation, focusing on eigenvalue degeneracies, symmetries, and eliminating redundant parameters to facilitate physical and mathematical applications.

Contribution

It provides a novel parametrization approach that explicitly accounts for degeneracies and symmetries in density matrices, simplifying their representation and analysis.

Findings

Degeneracies are characterized by pairs of identical eigenvalues.

All degeneracies are represented as phase-rotation blocks within the unitary matrix.

A diagrammatic method is proposed to transform phase configurations.

Abstract

This paper presents a parametrization of a degenerate density matrix. The problem needs to be approached first with a diagonalized form (the spectral representation) to deal with degeneracy. Such a form is useful for this parametrization in that the conditions to be a density matrix from a Hermitian matrix are applied only to a diagonal eigenvalue matrix, not a unitary matrix. Those conditions can be satisfied by parametrizing eigenvalues with squared spherical coordinates in dimension of the matrix. Degeneracy in eigenvalues brings symmetries between a eigenvalue matrix and a unitary matrix, which are realized in a form of a commuting unitary matrix, called a commutant. The associated redundant parameters in a unitary matrix have to be eliminated. It is realized in this paper that degrees of degeneracies can be defined as the total number of possible pairs of the same eigenvalues and one degree of degeneracy corresponds to one phase and one two dimensional rotation in a unitary matrix or a commutant. In this way all the degeneracies are identified and assigned to one phase-one rotation block. Therefore, a unitary matrix or a commutant is a product of these blocks and a general diagonal phase matrix. In physics a unitary matrix is often parametrized by rotation and phase matrices, called an angular representation here. There are many possible different phase configurations. It is often not a trivial matter whether a given parametrized unitary matrix is general. A simple diagram will be introduced to illustrate how to transform one phase configuration to another.