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

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

2003-05-26

arxiv.org/abs/quant-ph/0305156v1

J.Phys. A38 (2005) 2657-2668

Physics

Quantum Physics

21 pages

Scientific paper

10.1088/0305-4470/38/12/008

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.

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