%0 Journal Article %1 lami2016gaussian %A Lami, Ludovico %A Serafini, Alessio %A Adesso, Gerardo %D 2016 %K %T Gaussian entanglement revised %U http://arxiv.org/abs/1612.05215 %X 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.

@article{lami2016gaussian, abstract = {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.}, added-at = {2016-12-16T14:00:11.000+0100}, author = {Lami, Ludovico and Serafini, Alessio and Adesso, Gerardo}, biburl = {https://www.bibsonomy.org/bibtex/202d77e3f40eaf974dadf5b2b0f5150ce/klhamm}, description = {[1612.05215] Gaussian entanglement revised}, interhash = {0665ee66b3cdb3817e26b3d4ec7ab8aa}, intrahash = {02d77e3f40eaf974dadf5b2b0f5150ce}, keywords = {}, note = {cite arxiv:1612.05215Comment: 14 pages, 1 figure}, timestamp = {2016-12-16T14:00:11.000+0100}, title = {Gaussian entanglement revised}, url = {http://arxiv.org/abs/1612.05215}, year = 2016 }