## Appendix B Secret Key Rate in the Finite-Size Scenario  
In the following, let the notation \(N\) represent the total number of signals exchanged by [PERSON] and [PERSON] during the protocol. The \(x\) and \(y\) represent the classical data of [PERSON] and [PERSON] after they have measured their quantum states, and \(E\) refers to the quantum state of the eavesdropper. In the case of the CVQKD system, the secret key rate obtained for a finite size analysis can be given by  
\[K_{f}=\frac{n}{N}\left(\beta I(x:y)-S_{e_{PE}}(y:E)-\Delta(n)\right), \tag{17}\]  
where \(\frac{n}{N}\) represents \(n\) signals are used for the establishment of the key, out of the \(N\) signals exchanged. The notation \(S_{e_{PE}}(y:E)\) is defined as the maximum of the Holevo information compatible with the statistics except with probability \(e_{PE}\), where \(e_{PE}\) is the failure probability of parameter estimation. The parameter \(\Delta(n)\) is related to the security of the privacy amplification given by  
\[\Delta(n)=(2\text{dimH}_{\text{y}}+3)\sqrt{\frac{\log_{2}(2/\varepsilon)}{n}}+ \frac{2}{n}\log_{2}(1/e_{PE}), \tag{18}\]where \(\mathcal{H}_{y}\) is the Hilbert space corresponding to the variable \(y\) used in the raw key. Since the raw key is usually encoded on binary bits, we have \(\text{dimH}_{y}=2\). In addition, \(\tilde{e}\) is a smoothing parameter, and \(e_{PA}\) is the failure probability of the privacy amplification procedure. Both the smoothing parameter \(\tilde{e}\) and \(e_{PA}\) are intermediate parameters which should be optimized numerically. One needs to fix an overall security parameter \(e\) for the performance analysis of the CVQKD system. Compared with the unconditional security in the case of asymptotic scenario, the CVQKD system is limited to \(e\)-security in a finite-size setting. The parameter \(e\) corresponds to the failure probability of the whole protocol, meaning that the protocol is assured to performed as it is supposed to except with a probability at most \(e\). The failure probability \(e\) can be computed from the various parameters as follows  
\[e=e_{PE}+e_{EC}+\tilde{e}+e_{PA}, \tag{10}\]  
where \(e_{EC}\) is the probability that the reconciliation fails. The above-mentioned error probabilities can be set to \(e\approx e_{PE}=e_{EC}=\tilde{e}=e_{PA}=10^{-10}\).  
In the finite-size scenario, \(S_{e_{PE}}(y:E)\) needs to be calculated in parameter estimation procedure where one can find a covariance matrix \(\gamma_{AB_{2e_{PE}}}\), which minimizes the secret key rate with a probability of at least \(1-e_{PE}\). The estimation of \(\gamma_{AB_{2e_{PE}}}\) can be calculated by \(m=N-n\) couples of correlated variables \((x_{i},y_{i})_{i=1,\cdots,m}\) in the following form  
\[\gamma_{AB_{1}e_{PE}}=\left[\begin{array}{ll}(\tau V_{A}+1)\mathbb{I}&t_{min} Z_{N}^{\prime}\sigma_{z}\\ t_{min}Z_{N}^{\prime}\sigma_{z}&(t_{min}^{2}\tau V_{A}+\sigma_{max}^{2}) \mathbb{I}\end{array}\right], \tag{11}\]  
where \(t_{min}\) and \(\sigma_{max}^{2}\) correspond respectively to the minimal value of \(t\) and the maximal value of \(\sigma^{2}\) compatible with the sampled data, except with probability \(e_{PE}/2\). With the maximum likelihood estimators, we have  
\[\begin{array}{l}t_{min}\approx\sqrt{T}-z_{e_{PE}/2}\sqrt{\frac{1+T \tilde{e}}{mV_{A}}},\\ \sigma_{max}^{2}\approx 1+T\tilde{e}+z_{e_{PE}/2}\frac{(1+T\tilde{e}) \sqrt{2}}{\sqrt{m}},\end{array} \tag{12}\]  
where \(z_{e_{PE}/2}\) is such that \(1-\text{erf}(z_{e_{PE}/2}/\sqrt{2})=e_{PE}/2\) and \(erf\) is the error function given by  
\[erf(x)=\frac{2}{\sqrt{\pi}}\int_{0}^{x}\mathrm{e}^{-\mathrm{t}^{2}}\,\mathrm{ dt}. \tag{13}\]