Analytic solution of the separability criterion for continuous variable systems

TL;DR

This paper derives explicit analytic formulas for squeezing parameters that determine the separability of continuous variable Gaussian systems, clarifying the relationship between the separability and P-representation conditions.

Contribution

It provides the first explicit analytic solutions for the squeezing parameters that satisfy the separability criterion in continuous variable systems.

Findings

Explicit formulas for squeezing parameters are derived.

The equivalence between separability and P-representation conditions is established.

Implications for previous related works are discussed.

Abstract

By using the algebraic separability criterion of R. Simon, we present an explicit determination of squeezing parameters for which the P-representation condition saturates the $Sp(2,R)\otimes Sp(2,R)$ invariant separability condition for continuous variable two-party Gaussian systems. We thus give for the first time 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.