Finite-level systems, Hermitian operators, isometries, and a novel parameterization of Stiefel and Grassmann manifolds

Abstract

In this paper we obtain a description of the Hermitian operators acting on the Hilbert space $\C^n$, description which gives a complete solution to the over parameterization problem. More precisely we provide an explicit parameterization of arbitrary $n$-dimensional operators, operators that may be considered either as Hamiltonians, or density matrices for finite-level quantum systems. It is shown that the spectral multiplicities are encoded in a flag unitary matrix obtained as an ordered product of special unitary matrices, each one generated by a complex $n-k$-dimensional unit vector, $k=0,1,...,n-2$. As a byproduct, an alternative and simple parameterization of Stiefel and Grassmann manifolds is obtained.

21 pages

References in corpus (3)

• On the volume of the set of mixed entangled states
• Characterization of the Positivity of the Density Matrix in Terms of the Coherence Vector Representation
• Hall Normalization Constants for the Bures Volumes of the n-State Quantum Systems

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