forces are shown to have correlations of the form $$\begin{aligned}
\label{p17}\langle f_{a}(t')f^{*}_{a}(t)\rangle & = {A\over B}\left[2\zeta'^{2}+2\chi-(2\zeta'+\zeta)e^{-\theta}\right] \delta(t-t'),\end{aligned}$$ $$\begin{aligned}
\label{p18}\langle f_{b}(t')f^{*}_{b}(t)\rangle = 0,\end{aligned}$$ $$\begin{aligned}
\label{p19}\langle f_{b}(t')f_{a}(t)\rangle&=
 \frac{A}{2B}\left[\zeta'(1+\zeta'\zeta)+(2-\zeta'\zeta)e^{-\theta}\right]\delta(t-t'),\end{aligned}$$ $$\begin{aligned}
\label{p20}\langle f^{*}_{b}(t')f_{a}(t)\rangle=\langle f_{a}(t')f_{a}(t)\rangle
 =\langle f_{b}(t')f_{b}(t)\rangle = 0.\end{aligned}$$

Continuous variable entanglement
================================

In recent years, on the basis of different conditions and assumptions, several inseparability criteria for continuous variable product states have been proposed [@prl842722; @prl96050503; @pla2231]. One of these is related to a logarithmic negativity ($E_{N}$) [@job7577; @pra65032314; @pra70022318] which can be defined as [@pra70022318; @prl93063601; @jpb41215502] $$\begin{aligned}
\label{p22}E_{N}=\rm{max}[0,-\log_{2}V_{s}],\end{aligned}$$ where $V_{s}$ is the smallest eigenvalue of the symplectic matrix [@pra70022318; @prl93063601]. The inseparability condition associated with this approach is based on the well established fact that for the composite state to be separable to its constitute, the product density operator should have a positive partial transpose.

In light of this, upon solving the eigenvalue equations for the symplectic spectrum of the covariance matrix of the partially transposed density operator, the smallest eigenvalue is found to have a form $$\begin{aligned}
\label{p23}V_{s}=\left[\frac{\xi-\sqrt{\xi^{2}-\rm4det\Xi}}{2}\right]^{1/2},\end{aligned}$$ with $$\begin{aligned}
\label{p24}\xi= \rm{det}\sigma_{A}+det\sigma_{B}-2det\sigma_{AB},\end{aligned}$$ where $\sigma_{A}$ and $\sigma_{B}$ are the covariance matrices describing each modes separately while $\sigma_{AB}$ represents the intermodal correlations [@pra65032314].

In connection with the bimodal states involved, the $2X2$ block 