  This is a continuation of the expository article \cite{krp} with some new
remarks. Let $S_n$ denote the set of all Gaussian states in the complex Hilbert
space $L^2 (\mathbb{R}^n),$ $K_n$ the convex set of all momentum and position
covariance matrices of order $2n$ in Gaussian states and let $\mathcal{G}_n$ be
the group of all unitary operators in $L^2 (\mathbb{R}^n)$ conjugations by
which leave $S_n$ invariant. Here we prove the following results. $K_n$ is a
closed convex set for which a matrix $S$ is an extreme point if and only if
$S=\frac{1}{2} L^{T} L$ for some $L$ in the symplectic group $Sp (2n,
\mathbb{R}).$ Every element in $K_n$ is of the form $\frac{1}{2} (L^{T} L +
M^{T} M)$ for some $L,M$ in $Sp (2n, \mathbb{R}).$ Every Gaussian state in $L^2
(\mathbb{R}^n)$ can be purified to a Gaussian state in $L^2 (\mathbb{R}^{2n}).$
Any element $U$ in the group $\mathcal{G}_n$ is of the form $U = \lambda W
({\bm {\alpha}}) \Gamma (L)$ where $\lambda$ is a complex scalar of modulus
unity, ${\bm {\alpha}} \in \mathbb{C}^n,$ $L \in Sp (2n, \mathbb{R}),$ $W({\bm
{\alpha}})$ is the Weyl operator corresponding to ${\bm {\alpha}} $ and $\Gamma
(L)$ is a unitary operator which implements the Bogolioubov automorphism of the
Lie algebra generated by the canonical momentum and position observables
induced by the symplectic linear transformation $L.$
