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arXiv:0710.5039

On the separability criterion for continuous variable systems

Kazuo Fujikawa

Published Oct 26, 2007• 1 citation
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Abstract

We present an elementary and explicit proof of the separability criterion for continuous variable two-party Gaussian systems. Our proof is based on an elementary formulation of uncertainty relations and an explicit determination of squeezing parameters for which the P-representation condition saturates the $Sp(2,R)\otimes Sp(2,R)$ invariant separability condition. We thus give the explicit formulas of squeezing parameters, which establish the equivalence of the separability condition with the P-representation condition, in terms of the parameters of the standard form of the correlation matrix. Our proof is compared to the past proofs, and it is pointed out that the original proof of the P-representation by Duan, Giedke, Cirac and Zoller(DGCZ) is incomplete. A way to complete their proof is then shown. It is noted that both of the corrected proof of DGCZ and the proof of R. Simon are closely related to our explicit construction despite their quite different appearances.

DOI
10.48550/arxiv.0710.5039

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We present an elementary and explicit proof of the separability criterion for continuous variable two-party Gaussian systems. Our proof is based on an elementary formulation of uncertainty relations and an explicit determination of squeezing parameters for which the P-representation condition saturates the Sp(2,R)⊗Sp(2,R)Sp(2,R)\otimes Sp(2,R)Sp(2,R)⊗Sp(2,R) invariant separability condition. We thus give the explicit formulas of squeezing parameters, which establish the equivalence of the separability condition with the P-representation condition, in terms of the parameters of the standard form of the correlation matrix. Our proof is compared to the past proofs, and it is pointed out that the original proof of the P-representation by Duan, Giedke, Cirac and Zoller(DGCZ) is incomplete. A way to complete their proof is then shown. It is noted that both of the corrected proof of DGCZ and the proof of R. Simon are closely related to our explicit construction despite their quite different appearances.
By using the algebraic separability criterion of Simon, we present an explicit determination of squeezing parameters for which the PPP-representation condition saturates the $\text{Sp}(2,R)\ensuremath{\bigotimes}\text{Sp}(2,R)$ invariant separability condition for continuous variable two-party Gaussian systems. We thus give the explicit analytic formulas of squeezing parameters which establish the equivalence of the separability condition with the PPP-representation condition. The implications of our algebraic analysis on some of the past related works are discussed.
A general separability condition on the second moment (covariance matrix) for continuous-variable two-party systems is derived by an analysis analogous to the derivation of Kennard's uncertainty relation without referring to the non-negativity of the partially transposed density matrix. This separability criterion is generally more stringent than that used by Simon which is based on the non-negativity of partially transposed density matrix, and thus this criterion may be useful in the analysis of general continuous two-party systems. Another separability criterion used by Duan et al. is shown to be generally weaker than that of Simon. We thus have a hierarchy of separability criterions, but all these criterions when combined with suitable squeezing become equivalent at the boundary of the PPP-representation condition and thus turned out to be sufficient to analyze the separability of two-party Gaussian systems.
Duan et al (Duan L-M, Giedke G, Cirac J I and Zoller P 2000 Phys. Rev. Lett. 84 2722) and, independently, Simon (Simon R 2000 Phys. Rev. Lett. 84 2726) have recently found necessary and sufficient conditions for the separability (classical correlation) of the Gaussian two-party (continuous variable) states. Duan et al remark that their criterion is based on a `much stronger bound' on the total variance of a pair of Einstein-Podolsky-Rosen-type operators than is required simply by the uncertainty relation. Here, we seek to formalize and test this particular assertion in both classical and quantum-theoretic frameworks. We first attach to these states the classical a priori probability (Jeffreys' prior), proportional to the volume element of the Fisher information metric on the Riemannian manifold of Gaussian (quadrivariate normal) probability distributions. Then, numerical evidence indicates that more than 99% of the Gaussian two-party states do, in fact, meet the more stringent criterion for separability. We collaterally note that the prior probability assigned to the classical states, that is those having positive Glauber-Sudarshan P -representations, is less than 0.001%. We, then, seek to attach as a measure to the Gaussian two-party states the volume element of the associated (quantum-theoretic) Bures (minimal monotone) metric. Our several extensive analyses, then, persistently yield probabilities of separability and classicality that are, to very high orders of accuracy, unity and zero, respectively, so the two quite distinct (classical and quantum-theoretic) forms of analysis are rather remarkably consistent in their findings.
Currently available separability criteria for continuous-variable states are generally based on the covariance matrix of quadrature operators. The well-known separability criterion of Duan et al. [L. M. Duan et al., Phys. Rev. Lett. 84, 2722 (2000)] and Simon [R. Simon, Phys. Rev. Lett. 84, 2726 (2000)], for example, gives a necessary and sufficient condition for a two-mode Gaussian state to be separable, but leaves many entangled non-Gaussian states undetected. Here we introduce an improvement of this criterion that enables a stronger entanglement detection. The improved condition is based on the knowledge of an additional parameter, namely, the degree of Gaussianity, and exploits a connection with Gaussianity-bounded uncertainty relations [A. Mandilara and N. J. Cerf, Phys. Rev. A 86, 030102 (2012)]. We exhibit families of non-Gaussian entangled states whose entanglement remains undetected by the Duan-Simon criterion.
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.