[@GCW03][^1]. This means that we have the one-to-one relation: $$Q(x_1, x_2, y_1,y_2) = P(\gamma x_1, -\gamma x_2, y_1, y_2).$$

We now consider general $n$-mode states. Because of the correspondence above, applying a random transformation $R\otimes R$ with $R$ drawn with the Haar measure on $K(n)$ on the classical data represented by the distribution $P$ is equivalent to a symmetrization in phase-space corresponding to the application of random conjugate passive linear transformations over the $n$ modes of Alice and Bob. Noting $\mathcal{G}$ the group of passive linear transformations in phase-space, the state obtained after the symmetrization of $n$ copies of the state $\rho$ (i.e., for a collective attack) is: $$\label{symm_state}
\mathcal{S}(\rho^{\otimes n}) := \int_{U \in \mathcal{G}} (U \otimes U^*) \, \rho^{\otimes n}  \, (U \otimes U^*)^\dagger \mathrm{d}U,$$ where $\mathrm{d} U$ is the Haar measure over the group $\mathcal{G}$.

Sketch of the security proof
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The rest of the paper consists in analyzing the state $\mathcal{S}(\rho^{\otimes n})$, and in particular proving that it becomes approximately Gaussian under conditions on the second and fourth moment of $P(x_1,x_2,y_1,y_2)$ which are usually met in practical implementations of a CVQKD protocol. Since the three distributions $W$, $Q$ and $P$, equivalently describe the protocol, we choose here 