Journal
JOURNAL OF OPTICS B-QUANTUM AND SEMICLASSICAL OPTICS
Volume 2, Issue 4, Pages L19-L24Publisher
IOP PUBLISHING LTD
DOI: 10.1088/1464-4266/2/4/101
Keywords
Gaussian two-party states; continuous variable states; classical states; thermal squeezed states; Glauber-Sudarshan P-representation; separability; Peres-Horodecki criterion; classically correlated; quantum entanglement; Bures (minimal monotone) metric; Fisher information; Jeffreys' prior; quantum Jeffreys' prior; Schrodinger kernel
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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.
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