Gaussian entanglement revisited

Abstract

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.

v3: minor changes; close to published version; v2: 17 pages, 1 figure. Novel results reported in the newly added Sections 5 and 8. Exposition improved, missing citations added, and typo in the title corrected

References in corpus (11)

• Steering, Entanglement, Nonlocality, and the EPR Paradox
• Entanglement in continuous variable systems: Recent advances and current perspectives
• Quantification of Gaussian quantum steering
• Unitarily localizable entanglement of Gaussian states
• Mode-Wise Entanglement of Gaussian States
• Quantification and scaling of multipartite entanglement in continuous variable systems
• Schur complement inequalities for covariance matrices and monogamy of quantum correlations
• On symplectic eigenvalues of positive definite matrices
• Gaussian quantum marginal problem
• Genuine multipartite entanglement of symmmetric Gaussian states: Strong monogamy, unitary localization, scaling behavior, and molecular sharing structure
• Rank three bipartite entangled states are distillable

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• Differential Parametric Formalism for the Evolution of Gaussian States: Nonunitary Evolution and Invariant States
• General class of continuous variable entanglement criteria
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