## 3 Improvement via Clusterization
### Suppressing Fading Noise via Clusterization
Finally, the total composable finite-size rate is the packages' average given by  
\[R_{\text{com}}=\frac{1}{m}\sum_{i=1}^{m}R_{\text{com}}^{i}. \tag{13}\]  
In our manuscript, we assume that \(v_{s}=\)1 GHz and \(v_{f}=\)1 KHz. During a run of 1 s, \(N=10^{9}\) pulses were transmitted between [PERSON] and [PERSON], while the measurement data can be divided into \(m=1/v_{f}=1000\) packages based on the time windows. Correspondingly, the size of each package is \(n=10^{6}\). For the individual package scheme, the measurement data for parameter estimation in each package (\(r\cdot n<10^{6}\)) are significantly low, which introduces a composable finite-size effect. Therefore, we consider the clusterization scheme to balance the trade-off between the fading noise and the composable finite-size effect. We divide all the individual packages into \(C\) groups (clusters), whereas the transmittance of the packages in the \(k\)-th cluster satisfies \(\tau\in[\tau_{\text{min}}^{k},\tau_{\text{max}}^{k}]\). Here, \(\tau_{\text{min}}^{k}\) and \(\tau_{\text{max}}^{k}\) are the upper and lower bounds of the \(k\)-th clusters, respectively. We determine the values of \(\tau_{\text{min}}^{k}\) and \(\tau_{\text{max}}^{k}\) according to the total PDT for an optimal division, where each group has an almost equal package number. Finally,we perform a security analysis for each cluster independently, and the final composable finite-size rate is their average, given by  
\[R_{\text{com}}=\mathcal{P}_{\text{ec}}r\sum_{k=1}^{C}\mathcal{P}^{k}\left[R_{ \text{asy}}\left(\eta_{\text{min}}^{k},\tilde{n}_{\text{max}}^{k}\right)-\frac{ \Delta_{\text{aep}}}{\sqrt{r\cdot N/C}}+\frac{C\cdot\log_{2}\bigl{(}2e_{h}^{2} \epsilon_{\text{cor}}\bigr{)}}{r\cdot N}\right], \tag{14}\]  
where \(\mathcal{P}^{k}\) denotes the probability of the package falling in the \(k\)-th cluster, and \(\eta_{\text{min}}^{k}\) and \(\tilde{n}_{\text{max}}^{k}\) are the estimators of effective parameters in the \(k\)-th cluster. In detail, the covariance matrix of the \(k\)-th cluster can be expressed as  
\[\gamma_{AB_{1}}^{k}=\left[ \begin{array}{cc}\mu\text{I}&\sqrt{\eta^{k}(\mu^{2 }-1)}Z\\ \sqrt{\eta^{k}(\mu^{2}-1)}Z&\left[\eta^{k}(\mu-1)+2\tilde{n}_{\text{eff}}^{k}+ 1\right]\text{I}\end{array} \right], \tag{15}\]  
with the effective parameters  
\[\eta^{k}=\left[\int_{\tau_{\text{min}}^{k}}^{\tau_{\text{max}}^{k}}\sqrt{\tau} \mathcal{P}(\tau)d\tau\right]^{2}, \tag{16}\]  
\[\tilde{n}_{\text{eff}}^{k}=\frac{\int_{\tau_{\text{min}}^{k}}^{\tau_{\text{ max}}^{k}}\tau P(\tau)d\tau-\eta^{k}}{2}(\mu-1)+\tilde{n}. \tag{17}\]  
Note that each cluster retains fading noise, as the packages within each cluster have relatively similar but not identical transmittance. Fortunately, the optimal clusterization based on the total PDT can control the composable finite-size effects within a suitable range while reducing \(\tilde{n}_{\text{fad}}\). Moreover, the cluster number cannot be too large, as the case of \(C=1000\) is equivalent to the individual package scheme with a heavy composable finite-size effect. The optimal cluster number is discussed in the following section.