On the separability criterion for continuous variable systems

TL;DR

This paper provides an explicit, elementary proof of the continuous variable Gaussian systems' separability criterion, clarifying the relation between the P-representation and separability conditions with explicit formulas.

Contribution

It offers a new, explicit proof of the separability criterion for Gaussian states, including formulas for squeezing parameters, and clarifies the relation to the P-representation condition.

Findings

Explicit formulas for squeezing parameters in Gaussian states

Equivalence established between separability and P-representation conditions

Clarification and completion of previous proofs by DGCZ and Simon

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.