  An important theorem in Gaussian quantum information tells us that we can
diagonalise the covariance matrix of any Gaussian state via a symplectic
transformation. Whilst the diagonal form is easy to find, the process for
finding the diagonalising 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 diagonalising symplectic from certain
submatrix determinants, which could prove useful in Gaussian quantum
information.
