## 3 Improvement via Clusterization
### Suppressing Fading Noise via Clusterization  
As mentioned above, the fading noise, \(\bar{n}_{\mathrm{fad}}\), is related to the statistics \(\mathrm{Var}\big{(}\sqrt{\tau}\big{)}\). However, we cannot change the channel statistics but only the method of data post-processing. In a practical setting, the frequency of signal preparation and detection \(\
u_{s}\) (reached GHz in a recent study [34]) is much higher than the fluctuation frequency of the atmospheric channel \(\
u_{f}\) (usually \(\sim\)1 KHz [10]). In other words, for a small time window, each transmitted pulse can be considered as passing through a stable channel with constant transmittance. Therefore, we can divide the arrived \(N\) pulses into \(m\) individual packages based on time windows (i.e., each package has \(n=N/m\) pulses), and each package has no transmittance fluctuation. In detail, as shown in Figure 1b, by detecting the intensity of the multiplexing beacon, we can monitor the real-time transmittance and the total PDT as the reference for  
Figure 3: Distribution of fading noise versus the satellite altitude and zenith angle: (**a**) downlink scenario, (**b**) uplink scenario. package division [10; 35]. Then, we calculate the secret key rate of each package individually and the final result is their average. Since there is no transmittance fluctuation in each package, the fading noise is zero. However, there is a negative impact of the individual package scheme when considering composable finite-size effects [36]. In a practical setting, users do not know all the parameters of the protocol, so they need to use part of the measurement data for parameter estimation. The asymptotic key rate in Figure 2 assumes that users can use the quantum communication channel an infinite number of times, thus having enough measurement data for perfect parameter estimation. This is impossible in a practical setting, where users can only use the quantum channel a finite number of times. Here, we assume that [PERSON] knows the signal modulation \(\mu\), while [PERSON] monitors the quantum efficiency \(\eta_{\text{eff}}\), thus only \(\tau\) and \(\bar{n}\) need to be estimated. In the composable finite-size regime, the secret key rate of an individual package can be expressed as [36]  
\[R_{\text{com}}^{i}=\mathcal{P}_{\text{ec}}r\left[R_{\text{asy}}\Big{(}\tau_{ \text{min}}^{i},\bar{n}_{\text{max}}\Big{)}-\frac{\Delta_{\text{aep}}}{\sqrt{r \cdot n}}+\frac{\log_{2}\big{(}2\epsilon_{\text{h}}^{2}\epsilon_{\text{cor}} \big{)}}{r\cdot n}\right], \tag{9}\]  
with the higher-order terms  
\[\Delta_{\text{aep}}=4\log_{2}\Big{(}\sqrt{d}+2\Big{)}\sqrt{\log_{2}(2/\epsilon _{5}^{2})}, \tag{10}\]  
where \(i=1,\ldots,m\) is the label of an individual package, \(r\) is the proportion of measured data used for secret key generation, \(\mathcal{P}_{\text{ec}}\) is the success probability of error correction, \(d\) is digitalization, \(\epsilon_{\text{h}}\) is the hashing parameter, \(\epsilon_{\text{s}}\) is the smoothing parameter, and \(\epsilon_{\text{cor}}\) is the \(\epsilon\)-correctness, which bounds the probability that [PERSON]'s and [PERSON]'s local strings are different after error correction and successful verification of their hashes. The estimated values \(\tau_{\text{min}}^{i}\) and \(\bar{n}_{\text{max}}^{i}\) are calculated using the maximum-likelihood estimators, which yield the results  
\[\tau_{\text{min}}^{i}\simeq\tau^{i}-w\sqrt{\frac{2\tau^{i}+\tau^{i}(2\bar{n}+v _{\text{det}})/\sigma_{\text{x}}^{2}}{(1-r)n}}, \tag{11}\]  
\[\bar{n}_{\text{max}}\simeq\bar{n}+w\frac{2\bar{n}+v_{\text{det}}}{\sqrt{2(1-r )n}}, \tag{12}\]  
where \(w=\sqrt{2}\text{erf}^{-1}\big{(}1-2\epsilon_{\text{pe}}\big{)}\) with the failing probability of parameter estimation \(\epsilon_{\text{pe}}\) and inverse error function \(\text{erf}^{-1}\), and the detector shot-noise \(v_{\text{det}}=1\) (\(v_{\text{det}}=2\)) for homodyne (heterodyne) detectors.