# Noiseless Attenuation for Continuous-Variable Quantum Key Distribution over Ground-Satellite Uplink
## 4 Performance Analysis
### Asymptotic Analysis  
As shown in Figure 5a, it demonstrates the secret key rate \(K_{A}\) as a function of zenith angle \(\zeta\) for various satellite altitudes. The attenuation coefficient \(g\) is optimal for each zenith angle. In addition, the single-photon catalysis (SPC) protocol, which can be considered as another noiseless attenuation operation [34], is also plotted for comparison. We find that the NA-based scheme improves performance. The allowable zenith angle is improved by \(6^{\circ}\) and \(8^{\circ}\) at \(H=100\) km and \(H=120\) km, respectively. Note that the satellite-mediated CV-QKD system has a strong dependence on the instantaneous position between the moving satellite and the ground station. The effective time of a communication cycle (satellite orbits the earth) is limited to the period when the satellite passes above the ground station. Therefore, a larger allowable zenith angle ensures a longer effective communication time so, as a consequence, more total secret keys. Figure 5b reveals that the NA operation increases the effective time. The calculation is based on [PERSON]'s second law with the earth mass \(M=5.965\times 10^{24}\) kg and the earth radius \(r_{0}=6371\) km.  
Figure 5: Secret key rate as a function of (**a**) zenith angle and (**b**) communication time during a single communication cycle. The blue, red, and yellow lines represent the case of ZPC-based protocol, SPC-based protocol, and coherent-state protocol, respectively. Solid lines and dashed lines represent two situations using the satellite altitude \(H=100\) km and \(H=120\) km, respectively.  
Another observation in Figure 5 is that, at a small zenith angle, the bound of the secret key rate of the NA-based CV-QKD protocol coincident with that of the coherent-state protocol. As the zenith angle increases, so does the strength of turbulence, the improvement caused by the NA operation starts to appear. This phenomenon can be attributed to the optimization of attenuation coefficient. Figure 6 illustrates the optimal attenuation coefficient \(g\) varying with the zenith angle \(\zeta\) for various satellite altitudes. The optimal attenuation coefficient equals 1 for the case of small zenith angle (\(\zeta<3^{\circ}\) and \(\zeta<26^{\circ}\) for 100 km and 120 km, respectively), and then shows a downward trend with the increasing zenith angle. In other words, the protocol without NA operation may perform better than the NA-based one under weak turbulence effects, while the NA-based protocol can reach higher secret key rate when the turbulence effects become stronger. Moreover, the NA operation may be required upon higher satellite altitude at \(H=120\) km.  
In the following, we show the effect of the NA operation on the security of the CV-QKD system. In Figure 7, we plot the reconciliation-stricken mutual information \(\beta_{r}I_{AB}\) and Holevo bound \(\chi_{E}\) as a function of attenuation coefficient \(g\). Without loss of generality, we select \(\zeta=0^{\circ}\) and \(\zeta=32^{\circ}\) to represent small zenith angle case and large zenith angle case, respectively. In addition, the satellite altitude is set to 100 km. We find that the use of the NA operation acts a negative effect on mutual information, but at the same time reduces the Holevo bound. For \(\zeta=0^{\circ}\), the secret key rate, represented by the gap between the green line and purple line, keeps increasing as the growth attenuation coefficient. For a zenith angle of \(32^{\circ}\), the secret key rate reaches the maximum at \(g=0.834\), and then decreases with the growth of attenuation coefficient. In a word, the NA operation can improve the secret key rate under a large zenith angle by relatively reducing the amount of information available to Eve, which is accomplished by selecting an optimal attenuation coefficient.  
Figure 6: The optimal attenuation coefficient as a function of zenith angle for the satellite altitudes \(H=100\) km (blue line) and \(H=120\) km (yellow line).