# Noiseless Attenuation for Continuous-Variable Quantum Key Distribution over Ground-Satellite Uplink
## 2 Uplink CV-QKD  
For the convenience of security analysis, the uplink CV-QKD protocol is structured with the equivalent entanglement-based (EB) scheme [27]. As shown in Figure 1, Alice generates a two-mode squeezed vacuum (TMSV) state as an entangled resource, which can be represented by its covariance matrix  
\[\gamma_{AB}=\left[ \begin{array}{cc}V\Pi&\sqrt{V^{2}-1}\sigma_{z}\\ \sqrt{V^{2}-1}\sigma_{z}&V\Pi\end{array} \right], \tag{1}\]  
where \(\sigma_{z}=\text{diag}(1,-1)\), \(\Pi=\text{diag}(1,1)\) and \(V=(1+\lambda^{2})/(1-\lambda^{2})\) with squeezing parameter \(\lambda\). Then, [PERSON] holds mode A for heterodyne detection, while sending another mode B to [PERSON] over the insecure uplink, which is assumed to be controlled by a potential eavesdropper. Due to the random fading of the channel, the initial quantum state transfer to a mixture [28], whose composition depends on the random transmittance \(T\) together with channel excess noise \(\epsilon\). Here, we assume a fixed \(\epsilon\), and describe the random \(T\) by its probabilities distribution. Specifically, we divide the fading channel into \(K\) stable sub-channels of individuals, while each sub-channel has a fixed transmittance \(\{T_{j}\}_{j\in\{1,\cdots,K\}}\) with corresponding probability \(\{P_{j}\}_{j\in\{1,\cdots,K\}}\). Therefore, the arriving mixture shared by [PERSON] and [PERSON] is  
\[\gamma_{AB_{2}}=\left[ \begin{array}{cc}V\Pi&\langle\sqrt{T}\rangle\sqrt{V^{2 }-1}\sigma_{z}\\ \langle\sqrt{T}\rangle\sqrt{V^{2}-1}\sigma_{z}&\langle T\rangle(V+\chi_{\text {line}})\Pi\end{array} \right], \tag{2}\]where \(\langle T\rangle=\sum_{j=1}^{K}P_{j}T_{j}\), \(\langle\sqrt{T}\rangle=\sum_{j=1}^{K}P_{j}\sqrt{T_{j}}\), and \(\chi_{\text{line}}=1/\langle T\rangle-1+\epsilon\). Finally, by implementing quadrature measurement, classical information reconciliation, and privacy amplification, [PERSON] and [PERSON] can extract the secret keys. As shown in Equation (2), the statistical parameters \(\langle T\rangle\) and \(\langle\sqrt{T}\rangle\) represents the fading effects of uplink on quantum state. To describe these parameters, we apply the elliptic-beam model [29; 30], which provides accurate agreement with the experimental data under weak and strong turbulence. In this model, the receiving beam is assumed to be an elliptic deformation, which can be expressed by a random vector  
\[v=(x_{0},y_{0},W_{1},W_{2},\varphi), \tag{3}\]  
where \((x_{0},y_{0})\) denotes the beam-centroid coordinates at telescope plane, \(W_{1}\) and \(W_{2}\) are the semi-axis of arriving elliptic beam, and \(\varphi\) is the orientation angle between semi-axis \(W_{1}\) and the \(x\) axis. Based on the vector \(v\), the probability distribution of transmittance (PDT) is given by  
\[P(T)=\frac{2}{\pi}\int_{\mathbb{R}^{4}}\mathrm{d}^{4}v\int_{0}^{\frac{\pi}{2} }\mathrm{d}\varphi\rho_{G}(v;\mu,\Sigma)\delta(T-T(v)), \tag{4}\]  
where \(T(v)\) defines the relationship between \(T\) and \(v\), and \(\rho_{G}(v;\mu,\Sigma)\) is a Gaussian probability density of \(v\) with the mean value \(\mu\) and the covariance matrix \(\Sigma\). The elements of \(\mu\) and \(\Sigma\) in the uplink case are listed in Appendix A. Based on \(\mu\) and \(\Sigma\), we can derive the PDT of the uplink channel through the Monte Carlo method [29], and then obtain the statistical parameters of \(T\). Note that the simulation parameters of the work (as shown in Table 1) are the same as our previous work so that the numerical analysis of transmittance vs. zenith angle can refer to [21]. Subsequently, we demonstrate the performance of the uplink CV-QKD protocol. We consider the asymptotic regime and assume an ideal homodyne detection. The information reconciliation is the reverse reconciliation, since it is more robust against channel attenuation [31].