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Finite-level systems, Hermitian operators, isometries, and a novel parameterization of Stiefel and Grassmann manifolds
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 -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 -dimensional unit vector, . As a
byproduct, an alternative and simple parameterization of Stiefel and Grassmann
manifolds is obtained.Comment: 21 page