Compatibility conditions on local and global spectra for $n$-mode Gaussian states

Compatibility conditions between the (global) spectrum of an $n$-mode Gaussian state and the spectra of the individual modes are presented, making optimal use of beam splitter and (two-mode) squeezing transformations. An unexpected bye-product of our elementary approach is the result that every two-mode Gaussian state is uniquely determined, modulo local transformations, by its global spectrum and local spectra -- a property shared not even by a pair of qubits.