A composite parameterization of unitary groups, density matrices and subspaces

Unitary transformations and density matrices are central objects in quantum physics and various tasks require to introduce them in a parameterized form. In this paper we present a parameterization of the unitary group U(d) of arbitrary dimension d which is constructed in a composite way. We show explicitly how any element of U(d) can be composed of matrix exponential functions of generalized anti-symmetric σ-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 Cd with the minimal number of parameters. As an example it is 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).