Zusammenfassung
In this paper we present a novel approach to the problem of separability
versus entanglement of Gaussian quantum states of bosonic continuous variable
systems, as well as new proofs of closely related results. We first review 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 greatly simplified
proofs of all these results. In particular, we recover the PPT-separability
equivalence theorem for Gaussian states of $1$ vs $n$ modes, for arbitrary $n$.
Next, we provide a previously unknown extension of this equivalence, proving
that it is valid also for arbitrary Gaussian states of $m$ vs $n$ modes that
are symmetric under the exchange of any two modes belonging to one of the
parties. Finally, we include a new proof of the sufficiency of the PPT
criterion for separability of isotropic Gaussian states, not relying on the
mode-wise decomposition of pure Gaussian states. 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. While this paper may be seen as a divertissement enjoyable by both
the quantum optics and the matrix analysis communities, the tools adopted here
are likely to be useful for further applications in continuous variable quantum
information theory, beyond the separability problem.
Nutzer