Perturbation bounds for Williamson's symplectic normal form

Martin Idel, Sebastián Soto Gaona, Michael M. Wolf

Research output: Contribution to journalArticlepeer-review

14 Scopus citations

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 STS is always stable. Finally, we sketch a few applications of the results in quantum information theory.

Original languageEnglish
Pages (from-to)45-58
Number of pages14
JournalLinear Algebra and Its Applications
Volume525
DOIs
StatePublished - 15 Jul 2017

Keywords

  • Perturbation bounds
  • Symplectic eigenvalues
  • Williamson's normal form

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