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arXiv:1112.2252 · International Journal of Modern Physics B

EXPLICIT FORMULA OF THE SEPARABILITY CRITERION FOR CONTINUOUS VARIABLES SYSTEMS

Kazuo Fujikawa

Published Sep 18, 2012• 1 citation
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Abstract

A very explicit analytic formula of the separability criterion of two-party Gaussian systems is given. This formula is compared to the past formulation of the separability criterion of continuous variables two-party Gaussian systems.
DOI
10.1142/s0217979212430060

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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)⊗Sp(2,R)Sp(2,R)\otimes Sp(2,R)Sp(2,R)⊗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.
By using the algebraic separability criterion of Simon, we present an explicit determination of squeezing parameters for which the PPP-representation condition saturates the $\text{Sp}(2,R)\ensuremath{\bigotimes}\text{Sp}(2,R)$ invariant separability condition for continuous variable two-party Gaussian systems. We thus give the explicit analytic formulas of squeezing parameters which establish the equivalence of the separability condition with the PPP-representation condition. The implications of our algebraic analysis on some of the past related works are discussed.
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 PPP-representation condition and thus turned out to be sufficient to analyze the separability of two-party Gaussian systems.