of the security of CV QKD with postselection.

Acknowledgments
===============

I thank Antonio Acín, Frédéric Grosshans and Philippe Grangier for fruitful discussions. I acknowledge support from the European Union under the ERC Starting grant PERCENT.

Proof of Lemma $[1]$ (Lemma \[1\] from [@LG10b]) {#proof-three-variables}
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Since the probability distribution $P$ is being randomized under the action of the group $K(n) = O(2n,\mathbb{R}) \cap Sp(2n,\mathbb{R})$ to give $\tilde{P}$ defined as $$\tilde{P}({\bf x}, {\bf y}) = \int_{K(n)} P(R{\bf x}, R{\bf y}) \mathrm{d} R$$ where $\mathrm{d}R$ refers to the Haar measure on $K(n)$, one has: $$\tilde{P}(R{\bf x}, R{\bf y}) = \tilde{P}({\bf x}, {\bf y})$$ for any ${\bf x}, {\bf y} \in \mathbb{R}^{2n}$ and $R \in K(n)$.

We want to show that any function $\tilde{P}: \mathbb{R}^{2n} \times \mathbb{R}^{2n}$, such that $\tilde{P}(R{\bf x}, R{\bf y}) = \tilde{P}({\bf x}, {\bf y})$ for any transformation $R \in K(n)$ only depends on the three following parameters: $||{\bf x}||^2, ||{\bf  y}||^2, {\bf x} \cdot {\bf y}$.

Given any four vectors ${\bf x}, {\bf x'}, {\bf y}, {\bf y'} \in \mathbb{R}^{2n}$ such that $||{\bf x}||^2=||{\bf x'}||^2, ||{\bf  y}||^2=||{\bf  y'}||^2, {\bf x} \cdot {\bf y}={\bf x'} \cdot {\bf y'}$, it is sufficient to exhibit a transformation $R\in K(n)$ such that ${\bf 