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arXiv:0902.2059 · Physical Review A

Analytic solution of the separability criterion for continuous-variable systems

Kazuo Fujikawa

Published Mar 25, 2009• 9 citations
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Abstract

By using the algebraic separability criterion of Simon, we present an explicit determination of squeezing parameters for which the $P$-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 $P$-representation condition. The implications of our algebraic analysis on some of the past related works are discussed.

DOI
10.1103/physreva.79.032334

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By using the algebraic separability criterion of Simon, we present an explicit determination of squeezing parameters for which the PPP-representation condition saturates the Sp(2,R)\ensuremath⨂Sp(2,R)\text{Sp}(2,R)\ensuremath{\bigotimes}\text{Sp}(2,R)Sp(2,R)\ensuremath⨂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.
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.
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.