# Semi-Empirical Satellite-to-Ground Quantum Key Distribution Model for Realistic Receivers
## 4 Satellite-to-Ground QKD Analysis
### Satellite-QKD Model and Parameters Verification  
We used the QKD model from Ref. [28], with the difference that the link transmission was time-dependent. Therefore, transmission of a \(k\)-photon state through the communication channel described in Section 3 is given by:  
\[\eta_{k}=1-(1-\eta(t))^{k} \tag{8}\]  
for \(k=0,1,2,\dots\). The photon number in each weak coherent pulse of the satellite transmitter has a Poisson distribution [13]. Pulses with a frequency \(f\) and intensities of signal \(\mu\), decoy \(\
u\), and vacuum \(\lambda\)=0 states are emitted at random with probabilities \(p_{s}\), \(p_{d}\), and \(p_{v}\), respectively. Background noise \(Y_{0}\), which includes detector dark counts as well as additional background contributions, was considered to be 250 and 500 counts per second for ground stations with 300 mm and 600 mm apertures, respectively. Note that the values were \begin{table}
\begin{tabular}{c c c c c c} \hline \(\mathbf{D_{T}}\) (mm) & \(\epsilon\) & \(\mathbf{\eta_{opt}}\) & \(\mathbf{\eta_{det}}\) & \(\mathbf{\gamma}\) (Rad) & \(\varkappa\) \\ \hline
300 & 0.81 & 0.44 & 0.55 & \(10^{-5}\) & 0.23–0.41 \\
600 & 0.73 & 0.27 & 0.55 & \(10^{-5}\) & 0.23–0.41 \\ \hline \end{tabular}
\end{table} 
Table 2: Experimental setup parameters for modeling channel between satellite and ground stations [27; 37; 38]. Figure 4: Link losses calculated for real passes of the Micius satellite over Zvenigorod observatory, with max elevation angles of 32.5\({}^{\circ}\) (31 October 2021; black curves), 57.5\({}^{\circ}\) (9 March 2022; red curves) and 42\({}^{\circ}\) (23 March 2022; blue curves). The yellow curves represent theoretical zenith passage with a maximum elevation angle of 90\({}^{\circ}\). Solid lines exhibit the 600 mm telescope data; dashed lines—300 mm. The computations are carried out for elevation angles greater than 20\({}^{\circ}\). We assumed good weather conditions, that is, \(\varkappa=0.23\). assumed to be constant during communication and were obtained while tracking the switched-off satellite by the ground stations. Additionally, intrinsic error \(e_{det}\) caused by receiver measurement fidelity remained independent of distance and was set at 0.9% and 0.5% for the 300 mm and 600 mm ground stations, respectively. The values used to simulate our QKD setups are listed in Table 3.  
The sifted key rate \(R_{sift}\) and QBER were calculated from parameters of the optical link for the weak and vacuum decoy-state BB84 model provided by [PERSON] et al. [28]. The expressions of values for signal states are given by:  
\[R_{sift}=\frac{1}{2}fp_{s}[Y_{0}+1-e^{-\mu\eta(t)}], \tag{9}\]  
\[QBER=\frac{1}{2}fp_{s}[e_{0}Y_{0}+e_{det}(1-e^{-\mu\eta(t)})]/R_{sift}. \tag{10}\]  
Figure 5 presents the calculated \(R_{sift}\) and QBER of signal states during the satellite passages with peak elevation angles of 32.5\({}^{\circ}\) and 90\({}^{\circ}\). The sifted key rate has a strong dependence on the maximum elevation angle and changes non-uniformly over the communication time. We list the sifted key rate at a maximum value are 3.3 kbit s\({}^{-1}\) and 11.4 kbit s\({}^{-1}\) for the 300 mm ground station, and 7.2 kbit s\({}^{-1}\) and 25.3 kbit s\({}^{-1}\) for the 600 mm one. The minimal sifted key generation corresponds to the elevation angle of 20\({}^{\circ}\) and equals 1.4 kbit s\({}^{-1}\) and 3.1 kbit s\({}^{-1}\) for the ground stations with the 300 mm and 600 mm apertures, respectively. The QBER of signal states \(E_{\mu}\) is also different for the ground stations [28] and is limited to 3% for the 300 mm aperture ground station, and to 2.5% for the 600 mm aperture. Figure 5: Modelled sifted key rate \(R_{sift}\) and QBER of signal states during the satellite passage above a ground station on 31 October 2021 and during the zenith satellite passage. The key parameters of the model are listed in Table 3. The computations were carried out for clear weather conditions and for elevation angles greater than 20\({}^{\circ}\).