  This paper presents two realizations of linear quantum systems for covariance
assignment corresponding to pure Gaussian states. The first one is called a
cascade realization; given any covariance matrix corresponding to a pure
Gaussian state, we can construct a cascaded quantum system generating that
state. The second one is called a locally dissipative realization; given a
covariance matrix corresponding to a pure Gaussian state, if it satisfies
certain conditions, we can construct a linear quantum system that has only
local interactions with its environment and achieves the assigned covariance
matrix. Both realizations are illustrated by examples from quantum optics.
