Stiefel流形和Grassmann流形的高阶系统、Hermitian算子、等距和一种新的参数化
Finite-level systems, Hermitian operators, isometries, and a novel parameterization of Stiefel and Grassmann manifolds
摘要：
本文给出了作用在Hilbert空间$\C^n$上的Hermitian算子的一个描述，该描述给出了过参数化问题的一个完整的解决方案.更确切地说，我们提供了一个明确的参数化的任意$n$维运营商，运营商可以被认为是哈密顿量，或密度矩阵有限级量子系统。它示出的频谱的多重性编码在一个标志酉矩阵作为一个特殊的酉矩阵，每一个由一个复杂的$n-k$维单位向量，$k=0,1,...,n-2$生成的有序产品。作为副产品，一个替代和简单的参数化Stiefel和格拉斯曼流形。
英文摘要：
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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