the Gaussian distribution with the same first two moments as $P_v$, one can prove that the variational distance $\Delta$ between the two distributions is of the form (see Appendix \[BE\] for a more precise bound): $$\Delta := \iiint_{\mathbb{R}^3} \left|P_v(V)-P_G(V)) \right| \mathrm{d}V= O\left(\frac{1}{\sqrt{n}}\right).$$ The scaling law in $O(1/\sqrt{n})$ is generic for Berry-Esseen theorem and the constant factor depends on the covariance matrix of $[X,Y,Z]$, that is, on the moment of order 4 of the measurement results $(x,y)$.

In general, one could compute the secret key rate corresponding to a state described by the distribution $P_G$. Here, we will restrict ourselves to a very concrete scenario, that of a quantum channel acting like a Gaussian channel (like an optical fiber typically [^2]). We insist that this is not a new assumption since one can always compute the covariance matrix $\Sigma$ of the distribution $P_G$ above. However, in general, the quantum channel will be such that $\Sigma$ will be close (up to sampling errors) to the covariance matrix obtained for a Gaussian channel.

The limiting distribution $P_G$ is close to an i.i.d. Gaussian distribution in typical implementations
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In practice, the coherent states are sent through an optical fiber acting as a Gaussian 