in direct and reverse reconciliation, i.e., $K(Y|X)$ and $K(X|Y)$.

It is easy to check the existence of wide range of parameters for which these two rates are strictly positive, so that Alice and Bob can extract a secret key despite the absence of entanglement (at the input state $\rho_{Aa}
$ and, therefore, also at the output state $\rho_{AB}$). As an example, we may consider the maximum correlation value $g=\mu-1$ for the separable Gaussian state $\rho_{Aa}$, and we may take the large modulation limit $\mu
\rightarrow+\infty$, as typical in continuous variable QKD. In this case, we get the following asymptotical expression for Alice and Bob’s mutual information$$I(X,Y)\rightarrow\log_{2}\frac{\tau\mu}{1+3\tau+(1-\tau)\omega}+O(\mu^{-1}),$$ and the following asymptotical spectra$$\begin{aligned}
\nu_{\mathbf{E}}^{-} & \rightarrow(1-\tau)\mu+\tau\omega+O(\mu^{-1}), \\
\nu_{\mathbf{E}}^{+} & \rightarrow\omega+O(\mu^{-1}), \\
\nu_{\mathbf{E}|Y}^{+} & \rightarrow\frac{1+(1-\tau)\omega}{\tau}+O(\mu
^{-1}), \\
\nu_{\mathbf{E}|X}^{\pm} & \rightarrow\xi_{\pm}+O(\mu^{-1}),\end{aligned}$$ where $$\begin{aligned}
\xi_{\pm} & :=\frac{\sqrt{(\omega+3)^{2}+\tau^{2}(\omega-3)^{2}-2\tau
(\omega^{2}+7)}}{2} \\
& \pm\frac{(1-\tau)(\omega-3)}{2}.\end{aligned}$$ Then, using the expansion $h(x)\simeq\log_{2}(ex/2)+O(1/x)$ for large $x$, we can write the two asymptotical rates$$\begin{aligned}
K(Y|X) & =R(\tau,\omega)+h(\xi_{+})+h(\xi_{-}), \\
K(X|Y) & =R(\tau,\omega)+h\left[ \frac{1+(1-\tau)\omega}{\tau}\right] ,\end{aligned}$$ where we have introduced the common term$$R(\tau,\omega):=\log_{2}\frac{2\tau}{e(1-\tau)[1+3\tau+(1-\tau)\omega ]}-h(\omega).$$ As we can see from Fig. \[soglie\], there are wide regions of positivity for these rates.

![*Left panel*. Rate $K(Y|X)$ in direct reconciliation, as a function of channel transmissivity $\protect\tau$ and thermal 