Perturbation bounds for Williamson's symplectic normal form

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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.

MSC

15B48
15A42
15A18

Keywords

Williamson's normal form
Symplectic eigenvalues
Perturbation bounds

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