quantum channel, meaning that the data obtained by Alice and Bob follow a Gaussian distribution. In general, this does not simplify the security analysis since observing variables that *look* Gaussian does not mean that they indeed *are* Gaussian. Here, we will show that *looking* Gaussian is sufficient for the bound obtained through Berry-Esseen theorem to be useful. Let us consider the case where $$(x_i, y_i) \sim \mathcal{N}\left(
\begin{bmatrix}
0\\
0\\
\end{bmatrix},
\begin{bmatrix}
\langle x_i^2 \rangle & \langle x_i y_i \rangle \\
\langle x_i y_i \rangle & \langle y_i^2 \rangle \\
\end{bmatrix}
 \right).$$ Then, applying the symmetrization and using the results of Berry-Esseen theorem, one obtains that the symmetrized (normalized) distribution $P_v$ tends to a Gaussian distribution with covariance matrix $\Sigma_G$: $$\Sigma_G = \left[
\begin{smallmatrix}
3\langle x_i^2 \rangle^2 &  \langle x_i^2 \rangle \langle y_i^2 \rangle + 2 \langle x_i y_i \rangle^2 & 3 \langle x_i^2 \rangle \langle x_i y_i \rangle  \\
\langle x_i^2 \rangle \langle y_i^2 \rangle + 2 \langle x_i y_i \rangle^2 & 3\langle y_i^2 \rangle^2& 3 \langle y_i^2 \rangle \langle x_i y_i \rangle\\
3 \langle x_i^2 \rangle \langle x_i y_i \rangle & 3 \langle y_i^2 \rangle \langle x_i y_i \rangle & \langle x_i^2 y_i^2 \rangle\\
\end{smallmatrix}\right].$$ Unfortunately, however, in 