2016/12/31 by Ludovico Lami, Alessio Serafini, Gerardo Adesso · 4 citations
Computer Science · Physics and Astronomy · #Covariance #Covariance matrix #Equivalence (formal languages) #Gaussian #Gaussian function #Gaussian random field #Matrix (chemical analysis) #Peres–Horodecki criterion #Quantum Computing Algorithms and Architecture #Quantum Information and Cryptography #Quantum Mechanics and Applications #Quantum entanglement #quant-ph
paper · pdf · doi:10.1088/1367-2630/aaa654
published as New J. Phys. 20:023030, 2018 · 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
openalex created_date 2017/01/26 · openalex publication_date 2018/01/09 · arxiv created 2018/02/21 · arxiv updated 2018/02/22 · openalex updated_date 2026/08/06
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 versus 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 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: (i) Gaussian states of 1 versus 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 prove 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 versus 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.