Parametrizations of degenerate density matrices

TL;DR

This paper develops a new parametrization method for degenerate density matrices using homogeneous space techniques, avoiding Lie algebra complexities, and provides concrete examples.

Contribution

It introduces a novel parametrization approach for degenerate density matrices via sections of quotient spaces, sidestepping traditional Lie algebra methods.

Findings

Provides explicit parametrization of degenerate density matrices.

Constructs sections for natural projections in homogeneous spaces.

Includes two concrete examples demonstrating the method.

Abstract

It turns out that a parametrization of degenerate density matrices requires a parametrization of $\mathfrak{F}=U(n)/({U(k_1)\times U(k_2)\times \cdots \times U(k_m)})\quad n=k_1 +\cdots + k_m $ where $U(k)$ denotes the set of all unitary $k\times k$-matrices with complex entries. Unfortunately the parametrization of this quotient space is quite involved. Our solution does not rely on Lie algebra methods {directly,} but succeeds through the construction of suitable sections for natural projections, by using techniques from the theory of homogeneous spaces. We mention the relation to the Lie algebra back ground and conclude with two concrete examples.