Real Symmetric Matrices Larger Than a Hermitian Matrix: Applications to Gaussian Quantum Information Processing*

In this paper, we investigate the set of real symmetric matrices that are larger than a given Hermitian matrix in the semidefinite order, with particular attention to its minimal element. We explore the applications of this result in Gaussian quantum information processing. Furthermore, we propose potential research directions, especially the extension of this result to the set of real symmetric matrices that are larger than multiple given Hermitian matrices. These topics are of interest in both matrix theory and quantum information processing.