  We present a novel approach to the separability problem for Gaussian quantum
states of bosonic continuous variable systems. We derive a simplified necessary
and sufficient separability criterion for arbitrary Gaussian states of $m$ vs
$n$ modes, which relies on convex optimisation over marginal covariance
matrices on one subsystem only. We further revisit the currently known results
stating the equivalence between separability and positive partial transposition
(PPT) for specific classes of Gaussian states. Using matrix analysis techniques
such as Schur complements and matrix means, we provide a unified treatment and
compact proofs of all these results. In particular, we recover the
PPT-separability equivalence for: (i) Gaussian states of $1$ vs $n$ modes; and
(ii) isotropic Gaussian states. In passing, we also retrieve (iii) the recently
established equivalence between separability of a Gaussian state and and its
complete Gaussian extendability. Our techniques are then applied to progress
beyond the state of the art. We show that: (iv) Gaussian states that are
invariant under partial transposition are necessarily separable; (v) the PPT
criterion is necessary and sufficient for separability for Gaussian states of
$m$ vs $n$ modes that are symmetric under the exchange of any two modes
belonging to one of the parties; and (vi) Gaussian states which remain PPT
under passive optical operations can not be entangled by them either. This is
not a foregone conclusion per se (since Gaussian bound entangled states do
exist) and settles a question that had been left unanswered in the existing
literature on the subject. This paper, enjoyable by both the quantum optics and
the matrix analysis communities, overall delivers technical and conceptual
advances which are likely to be useful for further applications in continuous
variable quantum information theory, beyond the separability problem.
