channel, exactly as in the case of protocols without postselection. The symmetrization we introduce below will not induce an exact Gaussification, but only an approximate one. However, the quality of the Gaussification, characterized by the fidelity between $\mathcal{S}\left(\rho_{AB}^{\otimes n}\right)$ and $\rho_G^{\otimes n}$ will increase with $n$, and tend to 1 if some experimentally verifiable conditions (on the moment of order 4 of $\rho_{AB}$) are met.

The symmetrization we consider here was introduced in [@LKG09] where it was argued that it corresponds to the natural symmetry for protocols using a Gaussian modulation of coherent states. In the EB scenario, before they both perform their heterodyne measurements, Alice and Bob would apply random conjugate passive linear transformations over their $n$ modes. Once this is done, they apply the usual postselection protocol. This symmetrization can also be used in the PM scenario, and crucially, one can simply apply it to the *classical* data ${\bf x}$ and ${\bf y}$ of Alice and Bob. More concretely, in the PM scenario, Alice and Bob follow the standard scenario of sending coherent states and performing a heterodyne detection for Bob. Then, Alice draws a random transformation with the Haar measure on the group $K(n):=O(2n,\mathbb{R}) \cap Sp(2n,\mathbb{R})$ (isomorphic 