# Parameter Optimization Based BPNN of Atmosphere Continuous-Variable Quantum Key Distribution
## 2 Transmittance and Security Analysis
### Secret Key Rate in the Atmosphere Turbulence Channel  
For the sake of the CVQKD investigation in the atmospheric channel, we firstly make analysis centering around the key rate through the fading channel. The schematic diagram of discrete modulated CVQKD in the atmosphere turbulence channel is shown in Figure 3. First, [PERSON] discretely modulates the quantum signal and then dispatches it to [PERSON] via a fading channel whose feature is transmittance distribution \(T\). After taking over the quantum signal, [PERSON] conducts the coherent detection on the received signals and acquires the raw key data. A practical detector is featured by an efficiency \(\eta\) and a noise \(v_{el}\) on account of detector electronics.  
Here, the reachable secret key rate of atmospheric discrete modulation coherent state CVQKD is presented on the basis of the calculation results of the above sections. In the case of collective attack, assume that [PERSON] and [PERSON] apply the reverse reconciliation with reconciliation efficiency \(\beta\); the key rate can be given by [48]  
\[K=\beta I(a:b)-\chi(b:E). \tag{8}\]  
With the assumption of channel's transmittance \(\langle T\rangle,\langle\sqrt{T}\rangle\) and excess noise \(\varepsilon\), the covariance matrix of \(\rho_{AB}\) is shown as [49]  
\[\gamma_{AB}=\left(\begin{array}{cc}(V_{A}+1)\mathbf{I}&\langle\sqrt{T} \rangle Z\sigma_{z}\\ \langle\sqrt{T}\rangle Z\sigma_{z}&[\langle T\rangle(V_{A}+\varepsilon)+1] \mathbf{I}\end{array}\right), \tag{9}\]  
where \(Z=V_{A}(\xi_{0}^{3/2}\xi_{1}^{-1/2}+\xi_{1}^{3/2}\xi_{2}^{-1/2}+\xi_{2}^{3/2} \xi_{3}^{-1/2}+\xi_{3}^{3/2}\xi_{0}^{-1/2})\), with  
\[\xi_{0,2}=\frac{1}{2}\text{exp}(-0.5V_{A})[\cosh(-0.5V_{A})\pm \cos(-0.5V_{A})], \tag{10}\] \[\xi_{1,3}=\frac{1}{2}\text{exp}(-0.5V_{A})[\sinh(-0.5V_{A})\pm \sin(-0.5V_{A})].\]  
Figure 3: Schematic diagram of prepare-and-measure discrete modulated continuous variable quantum key distribution (CVQKD) in the atmosphere turbulence channel. [PERSON] discretely modulates the quantum signal, and [PERSON] performs the coherent detection on the received states. PM, phase modulator, BS, beamsplitter, and PBS, polarization beamsplitter.  
According to covariance matrix, \(\gamma_{AB}\), \(I\left(a\!:\!b\right)\) and \(\chi\left(b\!:\!E\right)\) can be acquired (see Appendix B). Figure 4 shows the secret key rate versus the transmittance and modulation variance. The results indicate that the maximum key rate can be obtained by adjusting the modulation variance. Therefore, the optimal modulation variance \(V_{A}\) can be derived by maximizing the key rate under diverse system conditions. The system conditions include the transmission distance \(L\), the Rytov variance \(\sigma_{R}^{2}\), the detector efficiency \(\eta\), and the electrical noise \(v_{el}\).  
In this section, we have analyzed the transmittance of the atmospheric channel under different turbulence intensity and calculate the secret key rate of discrete modulated CVQKD in the atmospheric channel. Through the above analysis, we conclude that adjusting the modulation variance can increase the secret key rate of the system.