Generalized Euler Angle Parameterization for U(N) with Applications to SU(N) Coset Volume Measures

TL;DR

This paper introduces a generalized Euler angle parameterization for U(N), enabling volume calculations for various cosets and deriving measures for N-dimensional quantum states, extending previous SU(N) work.

Contribution

It provides a new generalized Euler angle parameterization for U(N) and formulas for coset volumes and state measures, expanding on prior SU(N) parameterizations.

Findings

Derived volume formulas for U(N), CP(N), and other cosets.

Established measures for N-dimensional density matrices.

Extended Euler angle parameterization from SU(N) to U(N).

Abstract

In a previous paper (math-ph/0205016) an Euler angle parameterization for SU(N) was given. Here we present a generalized Euler angle parameterization for U(N). The formula for the calculation of the volume for U(N), CP(N) as well as other SU(N) and U(N) cosets will also be given. In addition, the mixed and pure state product measures for N-dimensional density matrices under this parameterization will also be derived.