Perturbation Bounds for Williamson's Symplectic Normal Form

TL;DR

This paper investigates how small changes in positive semidefinite matrices affect Williamson's symplectic diagonalization, providing bounds on stability and implications for quantum information theory.

Contribution

It establishes norm bounds for the stability of symplectic eigenvalues and the diagonalizing matrix under perturbations, extending understanding of Williamson's decomposition.

Findings

Norm bounds for symplectic eigenvalues stability

Stability of the diagonalizing matrix $S$ when the spectrum is nondegenerate

Applications in quantum information theory

Abstract

Given a real-valued positive semidefinite matrix, Williamson proved that it can be diagonalised using symplectic matrices. The corresponding diagonal values are known as the symplectic spectrum. This paper is concerned with the stability of Williamson's decomposition under perturbations. We provide norm bounds for the stability of the symplectic eigenvalues and prove that if $S$ diagonalises a given matrix $M$ to Williamson form, then $S$ is stable if the symplectic spectrum is nondegenerate and $S^TS$ is always stable. Finally, we sketch a few applications of the results in quantum information theory.