In a continuous-variable quantum key distribution (CV-QKD) protocol, which is based on heterodyne detection at the receiver, the application of a noiseless linear amplifier (NLA) on the received signal before the detection can be emulated by the post-selection of the detection outcome. Such a post-selection, which is also called a measurement-based NLA, requires a cut-off to produce a normalisable filter function. Increasing the cut-off with respect to the received signals results in a more faithful emulation of the NLA and nearly Gaussian output statistics at the cost of discarding more data. While recent works have shown the benefits of post-selection via an asymptotic security analysis, we undertake the first investigation of such a post-selection utilising a composable security proof in the realistic finite-size regime, where this trade-off is extremely relevant. We show that this form of post-selection can improve the secure range of a CV-QKD over lossy thermal channels if the finite block size is sufficiently large and that the optimal value for the filter cut-off is typically in the non-Gaussian regime. The relatively modest improvement in the finite-size regime as compared to the asymptotic case highlights the need for new tools to prove the security of non-Gaussian cryptographic protocols. These results also represent a quantitative assessment of a measurement-based NLA with an entangled-state input in both the Gaussian and non-Gaussian regime.
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