# Noiseless Attenuation for Continuous-Variable Quantum Key Distribution over Ground-Satellite Uplink
## 2 Uplink CV-QKD
The secret key rate can be derived from Equation (2) with the support of Gaussian state extremality theorem [27], which is given by  
\[K_{A}(\langle T\rangle,\langle\sqrt{T}\rangle,\epsilon)=\beta_{r}I_{AB}-\chi_{E}, \tag{5}\]  
where \(\beta_{r}\) denotes the reconciliation efficiency, \(I_{AB}\) is the Shanon mutual information, and \(\chi_{E}\) represents the Holevo bound. The calculations of \(I_{AB}\) and \(\chi_{E}\) are shown in Appendix B. Moreover, the angle-of-arrival fluctuation caused by wavefront distortion will introduce a coupling problem between spatial beam and fiber on receiver detection [32]. The relationship between the fiber coupling efficiency \(\varsigma\) and angle-of-arrival \(\beta_{a}\) can be expressed as [33]  
\[\varsigma=\int_{-0.5\delta_{\text{tor}}}^{0.5\delta_{\text{tor}}}\frac{1}{ \sqrt{2\pi\langle\beta_{a}^{2}\rangle}f}\exp(\frac{-l^{2}}{2f^{2}\langle\beta_ {a}^{2}\rangle})\mathrm{d}l, \tag{6}\]  
Figure 1: EB scheme of the NA-based CV-QKD in the uplink scenario. BS: beam splitter; \(T\): channel transmittance; \(\varepsilon\): channel excess noise; \(S_{1},S_{2}\): single-mode squeezed state. where \(f\) denotes the focal length, \(d_{\text{cor}}\) is the diameter of the fiber. \(\langle\beta_{a}^{2}\rangle\) represents the variance of \(\beta_{a}\), which is related to refraction index structure parameter \(C_{n}^{2}\). Details of \(\langle\beta_{a}^{2}\rangle\) in uplink case refer to Appendix C. Hence, the secret key rate in Equation (5) should be modified to  
\[K_{A}(\langle T\rangle,\langle\sqrt{T}\rangle,\epsilon)=\varsigma\times(\beta _{r}I_{AB}-\chi_{E}). \tag{7}\]  
The secret key rates \(K_{A}\) as a function of the zenith angle \(\zeta\) for the given satellite altitude \(H\) is shown in Figure 2, where the simulation parameters refer to Table 1 in Section 4. In addition, the performance of the downlink scenario, analyzing with the same parameters, is also plotted for comparison. Numerical analysis reveals the favorable of downlink for communication, while illustrating the urgency of an enhancement scheme for the uplink case. Note that the maximum secret key rate (\(\zeta=0^{\circ}\)) of the downlink at 100 km is lower than that of the uplink because the fiber coupling problem is more severe in the downlink scenario [21]. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Variable** & **Value** & **Description** & **Reference** \\ \hline \(\zeta\) & 0–60\({}^{\circ}\) & \(Z\)enith angle & \\ \(a\) & 1 m & Receiver telescope radius & [21] \\ f/2a & 1 & Focal ratio & [33] \\ \(d_{\text{cor}}\) & 10 \(\mu\)m & Fiber core diameter & [33] \\ \(W_{0}\) & 80 mm & Initial beam radius & [33] \\ \(V-1\) & 4 SNU & Modulation variance & [34] \\ \(\lambda_{0}\) & 800 nm & Wavelength & [21] \\ \(\hat{P}_{r}\) & 95\% & Reconciliation efficiency & [21] \\ \(\epsilon\) & 0.01 SNU & Excess noise & [35] \\ \(C_{n}^{2}\) & 1.12 \(\times 10^{-16}\) m\({}^{-2/3}\) & Refraction index structure parameter & [17] \\ \(n_{0}\) & 0.61 m\({}^{-3}\) & Mean number of scatterer particles & [17] \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The simulation parameters. Figure 2: Secret key rate vs. zenith angle for different scenarios: downlink (dashed line) and uplink (solid line). The blue, red, and yellow lines represent the satellite altitude of 100 km, 120 km, and 800 km, respectively.