to work with $P$, which is the one directly observable in the practical implementation of the protocol.

Our proof will consist of two steps. First, we show that the distribution $P$ describing the state after the symmetrization tends to an explicit limiting function, where the convergence speed is $O(1/\sqrt{n})$. While one could in principle stop at this point and directly compute the secret key rate that can be extracted from this state, we will focus on the experimentally relevant scenario where the quantum channel behaves approximately as a Gaussian channel. In this case, we can bound the distance between the limiting distribution describing the whole protocol and a Gaussian identically and independently distributed (i.i.d.) function (thus corresponding to a collective Gaussian attack for which the key rate is already known) with an error term of order $O(1/\sqrt{n})$. In such a practical scenario, one can therefore bound the distance between the actual state and the state corresponding to a Gaussian attack by an arbitrary small quantity. Taking $n$ large enough, the secret key rate of the symmetrized protocol is therefore identical to the secret key rate against collective Gaussian attacks.

Convergence to a limiting distribution
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To simplify the notations, we 