a practical protocol, Alice and Bob only have access to a finite-precision estimation of the covariance matrix, and the one they measure, $\Sigma_{\mathrm{est}}$, and that they use in Berry-Esseen theorem will slightly differ from the ideal one above $\Sigma_G$. The typical estimation error is of the order of $1/\sqrt{m}$ if $m$ samples are used in the procedure. Assuming that the estimation is performed with a (small) constant fraction of the total samples $n$, the typical error will be on the order of $1/\sqrt{n}$, which is comparable to the error term of the Berry-Esseen theorem. This then implies that the variational distance between the two final distributions is also of that order (see Appendix \[finite\] for details).

Security analysis of a CV QKD protocol with postselection
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Using the results above, the distribution $P$ (or equivalently the $Q$-function) of the state describing the symmetrized version of the QKD protocol is $1/\sqrt{n}$-close in variational distance to a Gaussian distribution. Moreover, in a practical scenario, this Gaussian distribution corresponds to that of an i.i.d. Gaussian state. If one can make the error $1/\sqrt{n}$ small enough, then the security of the symmetrized protocol against collective attacks is equivalent to that of the usual (non-symmetrized) 