Phase Factors in Singular Value Decomposition and Schmidt Decomposition

TL;DR

This paper explores the role of phase factors in the singular value decomposition and Schmidt decomposition, providing a clear method to handle phase ambiguities in these decompositions.

Contribution

It introduces a systematic approach to consistently incorporate phase factors in SVD and Schmidt decomposition, clarifying their uniqueness properties.

Findings

Phase factors in SVD are unique up to diagonal matrices of phases.

The product of phase-factor matrices in SVD is unique.

A simple three-step method for consistent SVD and Schmidt decomposition is summarized.

Abstract

In singular value decomposition (SVD) of a complex matrix A, the singular vectors or the eigenvectors of AA{\dag} and A{\dag}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.