Essentially All Gaussian Two-Party Quantum States are a priori Nonclassical but Classically Correlated

Abstract

Duan, Giedke, Cirac and Zoller (quant-ph/9908056) and, independently, Simon (quant-ph/9909044) 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 ninety-nine percent 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 one-thousandth of one percent. 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 apparently quite distinct (classical and quantum-theoretic) forms of analysis are rather remarkably consistent in their findings.