state *given it was postselected*. In principle, one could do this analysis with the experimental data obtained from the EB version of the protocol; but one cannot directly reconstruct this covariance matrix from the data observed in the actual PM version of the protocol. Indeed, the probabilistic map corresponding to a successful postselection is too complicated and one cannot expect to analyze its effect on general non-Gaussian states. For this reason, our only hope for a security proof seems to be to somehow enforce the Gaussianity of the state $\rho_{AB}$ Alice and Bob will use in their protocol. The idea is therefore to add an extra step to the usual protocol, that will make the state $\rho_{AB}^{\otimes n}$ more Gaussian. Let us note $\mathcal{S}$ the quantum map induced by this symmetrization.

It is now clear that if one had $\mathcal{S}\left(\rho_{AB}^{\otimes n}\right) = \rho_G^{\otimes n}$ where $\rho_G$ is the bipartite Gaussian state with the same covariance matrix as $\rho_{AB}$, then the security of the symmetrized protocol against general collective attacks would be identical to the security of the original protocol against Gaussian attacks. One could then compute the secret key rate, simply from the transmission and excess noise of the quantum 