Simultaneous symplectic reduction of quadratic forms into normal forms

Abstract: A fundamental result in symplectic linear algebra states that for a given positive semi-definite quadratic form on a symplectic space there exists a symplectic basis in which the quadratic form reduces to a normal form. The special case of the aforementioned result for positive definite quadratic forms is known as Williamson's theorem. In this work, we establish necessary and sufficient conditions on positive semi-definite quadratic forms on a symplectic space to be simultaneously reduced to their normal forms in a common symplectic basis. In particular, we establish conditions on $2n \times 2n$ real symmetric positive definite matrices to be simultaneously diagonalizable by a symplectic matrix in the sense of Williamson's theorem. We also discuss some applications of the main result in quantum information theory and statistical thermodynamics.