Gaussian entanglement revisited

We present a novel approach to the problem of separability versus entanglement in Gaussian quantum states of bosonic continuous variable systems, as well as a collection of closely related results. 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 multimode Gaussian states. Using techniques based on matrix analysis, such as Schur complements and matrix means, we then provide a unified treatment and compact proofs of all these results. In particular, we recover the PPT-separability equivalence for Gaussian states of $1$ vs $n$ modes, for arbitrary $n$. We then proceed to show the novel result that Gaussian states invariant under partial transposition are separable.
 Next, we provide a previously unknown extension of the PPT-separability equivalence to arbitrary Gaussian states of $m$ vs $n$ modes that are symmetric under the exchange of any two modes belonging to one of the parties. Further, we include a new proof of the sufficiency of the PPT criterion for separability of isotropic Gaussian states, not relying on their mode-wise decomposition. In passing, we also provide an alternative proof of the recently established equivalence between separability of an arbitrary Gaussian state and its complete extendability with Gaussian extensions. Finally, we prove that Gaussian states which remain PPT under passive optical operations cannot 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