# Noiseless Attenuation for Continuous-Variable Quantum Key Distribution over Ground-Satellite Uplink
## 4 Performance Analysis
### Finite-Size Regime  
In the asymptotic case mentioned above, the secure analysis is performed with the premise that the length of the data is infinite, which is impossible in practice. Therefore, considering the finite-size effect in security analysis is necessary for practical implementations [38; 39; 40]. In the finite-size regime, we need to estimate the unknown parameters, including the channel transmittance and channel noise. In practice, these parameters are estimated using finite-sized data samples so that the secret key rate will be reduced. Although the finite-size effect in fiber link has been well studied, the analysis in free space is still a subject to be focused on since the channel fluctuation has a significant negative impact on the parameter estimation. To compensate for imprecise estimation, a suppression method, which divides the overall data into several clusters for fluctuation suppression, has  
Figure 7: The reconciliation-stricken mutual information (green line) and Holevo bound (purple line) vs. attenuation coefficient for the zenith angles: (**a**) \(\zeta=0^{\circ}\); (**b**) \(\zeta=32^{\circ}\).  
been proposed [41]. However, it also increases the difficulty of the software and hardware at the receiver.  
In this section, we consider the case that overall data are included for security analysis. With the overall data, the covariance matrix \(\gamma_{AB_{2}}\) can be equivalently parameterized as originating from a stable link with fixed transmittance \(\eta=\langle\sqrt{T}\rangle^{2}\), which is given by  
\[\gamma_{AB_{2}}=\left[\begin{array}{cc}X\Pi&\sqrt{\eta}Y\sigma_{z}\\ \sqrt{\eta}Y\sigma_{z}&\eta(X-1)+\xi^{2}\end{array}\right], \tag{15}\]  
where \(\xi^{2}=(\langle T\rangle-\langle\sqrt{T}\rangle^{2})(X-1)+1+\langle T\rangle\epsilon\). Following the rewritten covariance matrix above, the calculation of the secret key rate in finite-size regime can be expressed as  
\[K_{F}=\varsigma\times\Big{\{}r[K_{A}(\eta_{\min},\xi_{\max}^{2})-\Delta(n)] \Big{\}}, \tag{16}\]  
where \(r\) is the proportion of data used for secret key rate generation, \(\Delta(n)\) is related to the security of the privacy amplification given by  
\[\Delta(n)=7\sqrt{\frac{\log_{2}(2/\bar{\epsilon})}{r\cdot N}}+\frac{2}{r\cdot N }\log_{2}(1/\epsilon_{PA}), \tag{17}\]  
where \(N\) represents the total exchanged signals, \(\bar{\epsilon}\) is the smoothing parameter, and \(e_{PA}\) is the failure probability of the privacy amplification procedure. Based on the derived results in [38], the value of \(\eta_{\min}\) and \(\xi_{\max}^{2}\) can be calculated by the maximum-likelihood estimators, which has the form  
\[(\sqrt{\eta})_{\min}\approx\sqrt{\eta}-Z_{e_{PE}/2}\sqrt{\frac{1+\eta\xi}{N(1- r)(V-1)}}, \tag{18}\]  
\[\xi_{\max}^{2}\approx 1+\eta\xi+Z_{e_{PE}/2}\frac{(1+\eta\xi)\sqrt{2}}{\sqrt{N(1- r)}}, \tag{19}\]  
where \(Z_{e_{PE}/2}=\sqrt{2}\text{erf}^{-1}(1-\epsilon_{PE})\) with the failing probability of parameter estimation \(\epsilon_{PE}\) and inverse error function \(\text{erf}^{-1}\). Here, erf represents the error function defined as  
\[\text{erf}=\frac{2}{\sqrt{\pi}}\int_{0}^{x}e^{-t^{2}}\text{d}t. \tag{20}\]  
Figure 8 shows the secret key rates \(K_{F}\) as a function of the zenith angle \(\zeta\) for various satellite altitudes, where the attenuation coefficient is optimal for each zenith angle. The parameters \(\overline{\epsilon}\), \(\epsilon_{PE}\), and \(\epsilon_{PA}\) are \(10^{-10}\), and half of the signals are used for parameter estimation (\(r=0.5\)). We find that the fewer signals exchanged, the more pronounced of finite-size effect. When the number of the exchanged signals is \(10^{6}\), the performance shows a big gap compared with other scenarios, especially in the case of \(H=120\) km (no secret key rate in that case). As the block length is increased, more signals can be used for parameter estimation and key extraction, so as a consequence, the secret key rate becomes closer to the asymptotic scenario.