# Noiseless Attenuation for Continuous-Variable Quantum Key Distribution over Ground-Satellite Uplink
## Appendix A The Elements of \(\mu\) and \(\Sigma\) in Equation (4)  
The vector \(\mu\) of Equation (4) has the form  
\[\mu=(\langle x_{0}\rangle,\langle y_{0}\rangle,\langle W_{1}^{2}\rangle,\langle W _{2}^{2}\rangle), \tag{1}\]  
where the elements \(\langle x_{0}\rangle=\langle y_{0}\rangle=0\). The covariance matrix \(\Sigma\) can be read as  
\[\Sigma=\left[\begin{array}{cccc}\langle x_{0}^{2}\rangle&0&0&0\\ 0&\langle y_{0}^{2}\rangle&0&0\\ 0&0&\langle\Delta W_{1}^{2}\rangle&\langle\Delta W_{1}\Delta W_{2}\rangle\\ 0&0&\langle\Delta W_{1}\Delta W_{2}\rangle&\langle\Delta W_{2}^{2}\rangle \end{array}\right]. \tag{2}\]  
The details of \(\mu\) and \(\Sigma\) refer to the supplemental material of [29]. In this paper, we derive the \(\mu\) and \(\Sigma\) of uplink scenario with the equivalent ground-to-satellite atmospheric parameters in [17] together with the permittivity fluctuation spectrum [42]  
\[\Phi(\kappa)=0.132C_{n}^{2}\mid\kappa\mid-\frac{1}{\pi}+\frac{2n_{0}}{\pi k^ {4}}\mid f_{0}(\kappa;d_{\text{scat}})\mid^{2}, \tag{3}\]  
where \(k=2\pi/\lambda_{0}\) is the wave number, \(\mid\kappa\mid\in[\kappa_{0},\kappa_{m}]\) with \(\kappa_{0}\in 1/L_{0}\) and \(\kappa_{m}\in 1/l_{0}\) (\(l_{0}\) denotes the inner scales of turbulence). \(f_{0}(\kappa;d_{\text{scat}})\) is the amplitude of the wave scattered from a separate particle, whose calculation method can be found in [43]. Note that we use the Mie scattering theory in this paper, as we consider the scattering from fog instead of rain [42]. The last two parameters of \(\mu\) in the updated model can be calculated as  
\[\langle W_{i}^{2}\rangle=\frac{4H^{2}\text{sec}^{2}(\xi)}{k^{2}W_{0}^{2}}[ \frac{\pi}{8}n_{0}W_{0}^{2}\text{fsec}(\zeta)+1.8C_{n}^{2}k^{2}W_{0}^{\frac{5} {5}}\text{fsec}(\zeta)+1], \tag{4}\]where \(\tilde{h}=20\) km denotes the altitude of the layer with prominent atmosphere effects. For covariance matrix \(\Sigma\), the elements become  
\[\langle x_{0}^{2}\rangle=\langle y_{0}^{2}\rangle=1.16\tilde{h}C_{n}^{2}W_{0}^{- \frac{1}{3}}H^{2}\text{sec}^{3}(\zeta), \tag{10}\]  
\[\langle\Delta W_{i}\Delta W_{j}\rangle=(2\delta_{ij}-0.8)\frac{11 \hbar C_{n}^{2}H^{4}\text{sec}^{5}(\zeta)}{k^{2}W_{0}^{\frac{7}{3}}}\] \[\times[1+\frac{\pi}{8}n_{0}W_{0}^{2}\tilde{h}\text{sec}(\zeta)]. \tag{11}\]  
More details of the downlink scenario based on the satellite-to-ground atmospheric parameters refer to our previous work [21].  
Based on the elements above, we can estimate the probability distribution of transmittance (PDT) via the Monte Carlo method. In general, the transmittance \(T\) can be estimated by  
\[T=T_{0}\exp\left\{-\left[\frac{d/a}{R(\frac{2}{W_{\text{eff}}(\varphi-\varphi) })}\right]^{Q\left(\frac{2}{W_{\text{eff}}(\varphi-\varphi)}\right)}\right\}, \tag{12}\]  
where \(d=\sqrt{x_{0}^{2}+y_{0}^{2}}\) with the corresponding polar coordinates \((d,\phi)\), \(W_{\text{eff}}(\cdot)\) is the effective spot-radius with deformation effect, \(T_{0}\) is the trancemititance for the centered beam which can be calculated by  
\[T_{0}=1-I_{0}\left(a^{2}\left[\frac{1}{W_{1}^{2}}-\frac{1}{W_{2} ^{2}}\right]\right)\exp\left[-a^{2}(\frac{1}{W_{1}^{2}}+\frac{1}{W_{2}^{2}})\right]\] \[-2\left\{1-\exp\left[-\frac{a^{2}}{2}(\frac{1}{W_{1}}-\frac{1}{W_ {2}})^{2}\right]\right\}\] \[\times\exp\left\{-\left[\frac{(W_{1}+W_{2})^{2}}{|W_{1}^{2}-W_{2 }^{2}|}\right]^{\lambda(\frac{1}{W_{1}}-\frac{1}{W_{2}})}\right\}, \tag{13}\]  
where \(R(\cdot)\) is the scale functions, \(\lambda(\cdot)\) is the shape functions. The two functions can be expressed as  
\[R(\xi)=\left[\ln\left(2\frac{1-\exp[-\frac{1}{2}a^{2}\xi^{2}]}{1-\exp[-a^{2} \xi^{2}]I_{0}(a^{2}\xi^{2})}\right)\right]^{-\frac{1}{\lambda(\xi)}}, \tag{14}\]  
\[\lambda(\xi)=2a^{2}\xi^{2}\frac{\exp(-a^{2}\xi^{2})I_{1}(a^{2}\xi^{2})}{1-\exp (-a^{2}\xi^{2})I_{0}(a^{2}\xi^{2})}\] \[\times\left[\ln\left(2\frac{1-\exp[-\frac{1}{2}a^{2}\xi^{2}]}{1- \exp[-a^{2}\xi^{2}]I_{0}(a^{2}\xi^{2})}\right)\right]^{-1}, \tag{15}\]  
where \(I_{i}(\cdot)\) is the modified Bessel function of the \(i\)-th order.