Perturbation Bounds for Williamson's Symplectic Normal Form

2016/09/05 by Martin Idel, Idel, Martin, Sebatian Soto Gaona +3
Mathematics · Physics and Astronomy · #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Quantum Physics (quant-ph) #Spectral Theory (math.SP) #math-ph #math.MP #math.SP #quant-ph

paper · pdf · doi:10.48550/arxiv.1609.01338

14+3 pages

arxiv created 2016/09/05 · arxiv updated 2016/09/07

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.

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