A composite parameterization of unitary groups, density matrices and subspaces

TL;DR

This paper introduces a composite parameterization of the unitary group, density matrices, and subspaces that simplifies the identification of redundant parameters, aiding quantum state analysis and entanglement measures.

Contribution

It presents a new composite parameterization of unitary groups and density matrices that reduces redundancy and simplifies the analysis of quantum states and subspaces.

Findings

Parameterization explicitly constructed using matrix exponentials of generalized anti-symmetric matrices.

Redundancy-free density matrices of arbitrary rank formulated.

Parameter reduction improves analysis of quantum state distillability.

Abstract

Unitary transformations and density matrices are central objects in quantum physics and various tasks require to introduce them in a parameterized form. In the present article we present a parameterization of the unitary group $\mathcal{U}(d)$ of arbitrary dimension $d$ which is constructed in a composite way. We show explicitly how any element of $\mathcal{U}(d)$ can be composed of matrix exponential functions of generalized anti-symmetric $\sigma$-matrices and one-dimensional projectors. The specific form makes it considerably easy to identify and discard redundant parameters in several cases. In this way, redundancy-free density matrices of arbitrary rank $k$ can be formulated. Our construction can also be used to derive an orthonormal basis of any $k$-dimensional subspaces of $\mathbb{C}^d$ with the minimal number of parameters. As an example it will be shown that this feature leads to a significant reduction of parameters in the case of investigating distillability of quantum states via lower bounds of an entanglement measure (the $m$-concurrence).