# Noiseless Attenuation for Continuous-Variable Quantum Key Distribution over Ground-Satellite Uplink
## Appendix D Derivation of \(\gamma_{AB_{1}}\)  
From [27], the covariance matrix \(\gamma_{AB_{1}}\) can be expressed as follows  
\[\gamma_{AB_{1}}=\begin{bmatrix}\langle\hat{q}_{1}^{2}\rangle&0&\langle\hat{q} _{1}\hat{q}_{2}\rangle&0\\ 0&\langle\hat{p}_{1}^{2}\rangle&0&\langle\hat{p}_{1}\hat{p}_{2}\rangle\\ \langle\hat{q}_{1}\hat{q}_{2}\rangle&0&\langle\hat{q}_{2}^{2}\rangle&0\\ 0&\langle\hat{p}_{1}\hat{p}_{2}\rangle&0&\langle\hat{p}_{2}^{2}\rangle\end{bmatrix}. \tag{114}\]  
We need to calculate the six mean values in Equation (114). Here we first introduce the input-output relation based on the characteristic function (CF). Then, the covariance matrix can be derived by using the relation between the mean values of observables and the CF. First of all, the CF of TMSV is given by [46]  
\[\chi_{\text{in}}(\alpha,\beta)=e^{-\frac{V}{2}\left(|\alpha|^{2}+|\beta|^{2} \right)+\frac{\sqrt{\gamma^{2}-1}}{2}(\alpha\beta+\alpha^{*}\beta^{*})}, \tag{115}\]  
The input-output relation of the ZPC operation can be theoretically expressed as  
\[\rho_{\text{out},b}=\text{Tr}_{b^{\prime}}[B\rho_{\text{in},b}|\ 0\rangle_{b^{ \prime}b^{\prime}}\langle 0\ |\ B^{\dagger}\ |\ 0\rangle_{b^{\prime}b^{\prime}}\langle 0\ |], \tag{116}\]  
where B denotes the beam splitter operator. Therefore, the CF corresponding to \(\rho_{\text{out},b}\) has the form  
\[\chi_{\text{out}}(\beta)=\int\frac{\text{d}^{2}\gamma}{\pi R}\chi_{\text{in}}( \gamma)\chi_{0}(\gamma_{1})\chi_{0}(\gamma_{2}), \tag{117}\]  
where \(\gamma_{1}=\beta/\sqrt{R}-\gamma\sqrt{T_{A}/R}\), \(\gamma_{2}=\gamma/\sqrt{R}-\beta\sqrt{T_{A}/R}\), \(\chi_{\text{in}}\) is the CF of the input state, \(\chi_{0}\) is the CF of the zero-photon Fock state with  
\[\chi_{0}(\gamma_{1})=e^{-\frac{1}{2}|\gamma_{1}|^{2}}L_{0}\Big{(}|\ \gamma_{1}\ |^{2}\Big{)}, \tag{118}\]  
where \(L_{0}(x)\) are the Laguerre polynomials. Here we use the Weyl expansion of the density operator, which has the form  
\[\rho=\int_{-\infty}^{\infty}\frac{\text{d}^{2}\gamma}{\pi}\chi(\gamma)D(- \gamma), \tag{119}\]  
where \(D(\gamma)\) is the displaced operator. Finally, the CF after the ZPC operation becomes  
\[\chi(\alpha,\beta)=\frac{1}{F_{0,0}(0,0)}\chi_{\text{in},\bar{\lambda}}( \alpha,\beta)F_{0,0}(\alpha,\beta), \tag{120}\]  
where \(\chi_{\text{in},\bar{\lambda}}(\alpha,\beta)\) is the CF of the TMSV with a modified squeezing parameter \(\bar{\lambda}=\lambda\sqrt{T}\), \(F_{0,0}(\alpha,\beta)\) is the non-Gaussian term resulting from non-Gaussian operation given by  
\[F_{0,0}(\alpha,\beta)=(1-\lambda^{2})/(1-\bar{\lambda}^{2}). \tag{121}\]In terms of the relation between the mean values of observables and the CF [47], the mean values in Equation (A26) can be calculated as  
\[\left\langle\hat{q}_{1}^{2}\right\rangle=-\frac{\partial^{2}}{\partial p_{1}^{2} }\chi(0,p_{1};0,0)|_{p_{1}=0},\] (A34)  
\[\left\langle\hat{p}_{1}^{2}\right\rangle=-\frac{\partial^{2}}{\partial q_{1}^{2 }}\chi(q_{1},0;0,0)|_{q_{1}=0},\] (A35)  
\[\left\langle\hat{q}_{1}\hat{q}_{2}\right\rangle=-\frac{\partial^{2}}{\partial p _{1}\partial p_{2}}\chi(0,p_{1};0,p_{2})|_{p_{1},p_{2}=0},\] (A36)  
\[\left\langle\hat{p}_{1}\hat{p}_{2}\right\rangle=-\frac{\partial^{2}}{\partial q _{1}\partial q_{2}}\chi(q_{1},0;q_{2},0)|_{q_{1},q_{2}=0},\] (A37)  
where \(\chi(\alpha,\beta)=\chi(q_{1},p_{1};q_{2},p_{2})\) with \(\alpha=q_{1}+ip_{1}\) and \(\beta=q_{2}+ip_{2}\).