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.
Symplectic decomposition from submatrix determinants / Pereira J.L.; Banchi L.; Pirandola S.. - In: PROCEEDINGS OF THE ROYAL SOCIETY OF LONDON. SERIES A. - ISSN 1364-5021. - ELETTRONICO. - 477:(2021), pp. 20210513-20210523. [10.1098/rspa.2021.0513]