A composite parameterization of unitary groups, density matrices and subspaces

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).