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arXiv:quant-ph/9909062 · Journal of Optics B Quantum and Semiclassical Optics

Essentially all Gaussian two-party quantum states area priorinonclassical but classically correlated

Paul B. Slater

Published Jul 13, 2000• 11 citations

Abstract

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.

DOI: 10.1088/1464-4266/2/4/101

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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.

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.

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

P

-representation condition and thus turned out to be sufficient to analyze the separability of two-party Gaussian systems.

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.

Abstract We investigate the separability of the two-mode Gaussian states (TMGSs) by using the variances of a pair of Einstein–Podolsky–Rosen (EPR)-like observables. Our starting point is inspired by the general necessary condition of separability introduced by Duan et al (2000 Phys. Rev. Lett . 84 2722). We evaluate the minima of the normalized forms of both the product and sum of such variances, as well as that of a regularized sum. Making use of Simon’s separability criterion, which is based on the condition of positivity of the partial transpose (PPT) of the density matrix (Simon 2000 Phys. Rev. Lett . 84 2726), we prove that these minima are separability indicators in their own right. They appear to quantify the greatest amount of EPR-like correlations that can be created in a TMGS by means of local operations. Furthermore, we reconsider the EPR-like approach to the separability of TMGSs which was developed by Duan et al with no reference to the PPT condition. By optimizing the regularized form of their EPR-like uncertainty sum, we derive a separability indicator for any TMGS. We prove that the corresponding EPR-like condition of separability is manifestly equivalent to Simon’s PPT one. The consistency of these two distinct approaches (EPR-like and PPT) affords a better understanding of the examined separability problem, whose explicit solution found long ago by Simon covers all situations of interest.

Certification and quantification of correlations for multipartite states of quantum systems appear to be a central task in quantum information theory. We give here a unitary quantum-mechanical perspective of both entanglement and Einstein-Podolsky-Rosen (EPR) steering of continuous-variable multimode states. This originates in the Heisenberg uncertainty relations for the canonical quadrature operators of the modes. Correlations of two-party

(N\phantom{\rule{0.16em}{0ex}}\text{vs}\phantom{\rule{0.16em}{0ex}}1)

-mode states are examined by using the variances of a pair of suitable EPR-like observables. It turns out that the uncertainty sum of these nonlocal variables is bounded from below by local uncertainties and is strengthened differently for separable states and for each one-way unsteerable state. The analysis of the minimal properly normalized sums of these variances yields necessary conditions of separability and EPR unsteerability of

(N\phantom{\rule{0.16em}{0ex}}\text{vs}\phantom{\rule{0.16em}{0ex}}1)

-mode states in both possible ways of steering. When the states and the performed measurements are Gaussian, then these conditions are precisely the previously known criteria of separability and one-way unsteerability.