论文标题
奇异值分解和施密特分解的相位因子
Phase Factors in Singular Value Decomposition and Schmidt Decomposition
论文作者
论文摘要
在复杂矩阵A的奇异值分解（SVD）中，AA†和A†a的奇异向量或特征向量是独特的。因此，SVD中的两个统一矩阵是独特的，直到相位因子的对角线矩阵，即相位因子矩阵。同样，这两个相位矩阵的乘积或相同奇异值的相应奇异向量的相因的乘积是独特的。在Schmidt分解中，相位因子矩阵是仅作用于子系统的相位旋转算子。我们在这里总结了三个简单的步骤，以始终如一地执行SVD和Schmidt分解，包括相位因子。
In singular value decomposition (SVD) of a complex matrix A, the singular vectors or the eigenvectors of AA† and A†A are unique up to complex phase factors. Thus, the two unitary matrices in SVD are unique up to diagonal matrices of phase factors, the phase-factor matrices. Also, the product of these two phase-factor matrices, or the product of phase factors of the corresponding singular vectors with the same singular value, is unique. In the Schmidt decomposition, a phase-factor matrix is a phase rotation operator acting on a subsystem alone. We summarize here three simple steps to consistently carry out the SVD and the Schmidt decomposition including the phase factors.
