Symplectic decomposition from submatrix determinants. (24th November 2021)

• Record Type: Journal Article
• Authors: Pereira, Jason L.; Banchi, Leonardo; Pirandola, Stefano
• Is Part Of: Proceedings. Volume 477:Number 2255(2021)
• Publication Date: 2021-11-24
• Subjects: Williamson decomposition -- symplectic matrices -- Gaussian quantum information; Physical sciences -- Periodicals; Engineering -- Periodicals; Mathematics -- Periodicals; 500
• DOI: 10.1098/rspa.2021.0513
• Languages: English
• ISSNs: 1364-5021

Abstract

Abstract : An important theorem in Gaussian quantum information tells us that we can diagonalize the covariance matrix of any Gaussian state via a symplectic transformation. While the diagonal form is easy to find, the process for finding the diagonalizing symplectic can be more difficult, and a common, existing method requires taking matrix powers, which can be demanding analytically. Inspired by a recently presented technique for finding the eigenvectors of a Hermitian matrix from certain submatrix eigenvalues, we derive a similar method for finding the diagonalizing symplectic from certain submatrix determinants, which could prove useful in Gaussian quantum information.