# Finite-Key Analysis of 1-Decoy Method Quantum Key Distribution with Intensity Fluctuation
## 3 Numerical Simulation  
In this section, we simulate a fiber-based QKD system model from [21]. The parameters, namely the intensities \(\mu_{i}\), the relative probability and the probability of choosing the \(Z\) basis, are optimized for different channel attenuation. The system operates at a repetition rate of 1 GHz. For the detectors, we assume that the dead-time \(t_{dead}\) is 100 ns and the dark count rate \(p_{d}=10^{-8}\). In addition, the optical misalignment error rate \(p_{per}\) is fixed to 0.001. The efficiency of error correction \(f_{EC}\) is 1.16. The experimental parameters are shown in Table 1.  
Based on the parameters above, we simulate the secret key rate curve. We firstly simulate the effect of data size on the security key rate, i.e., \(N_{Z}=10^{8},10^{9},10^{10},10^{20}\) for \(\delta_{\mu}=0,0.09\). As shown in Figure 1, when the data size is small, the secret key rate is greatly affected by the fluctuation in intensity. The main reason for this impact is that the larger the amount of data, the less sensitive it is to statistical fluctuations. From Equations (5) and (6), we can infer that a large data size can estimate \(N_{\alpha,\mu_{i}}^{*}\) and \(M_{\alpha,\mu_{i}}^{*}\) more closely. Therefore, we can achieve a tighter secure key rate in the later calculation. As the data size increases to a certain extent, its impact on statistical fluctuations will become less obvious and the curves of the secure key rate will converge. In Figure 1 right, the key rate curve decreases rapidly compared with the case without intensity fluctuations when \(N_{Z}=10^{8}\) and \(\delta_{\mu}=0.09\). Intensity fluctuation parameter \(\delta_{\mu}\) provides us with the range of intensity and helps us to estimate the secure key rate considering the worst case. However, when the data size tends to infinity, the key rate curve is less affected by the intensity fluctuations. Then, we simulate the effects of intensity fluctuations in different degrees on the 1-decoy state method. Assuming the secret key rates with degrees of intensity fluctuations, i.e., \(\delta_{\mu}=0,0.01,0.05,0.09\), for \(N_{Z}=10^{8}\) and \(N_{Z}=10^{9}\), respectively. We can see the results in Figure 2. Obviously, the intensity fluctuations have non-negligible effects on the secret key \begin{table}
\begin{tabular}{c c c} \hline
**Variable** & **Parameter** & **Simulation Value** \\ \hline \(p_{d}\) & Dark count rate & \(10^{-8}\) \\ \(p_{per}\) & Optical misalignment error & 0.01 \\ \(t_{dead}\) & Dead time & 100 ns \\ \(\epsilon_{sec}\) & Secrecy parameter & \(10^{-9}\) \\ \(\epsilon_{cor}\) & Correctness parameter & \(10^{-15}\) \\ \(f_{EC}\) & Efficiency of error correction & 1.16 \\ \hline \end{tabular}
\end{table} 
Table 1: Simulation parameters [21]. rate; the transmission distance is significantly reduced with the increase in the intensity fluctuation parameter. As shown in Equations (9), (22), (24) and (25), the counts \(s_{Z,0}^{L},s_{Z,1}^{L}\) and the phase error \(\phi_{Z}^{U}\) are all affected by strength fluctuations, leading to a reduction in the secure key rate. Moreover, the effects of intensity fluctuations are more obvious for \(N_{Z}=10^{8}\) than \(N_{Z}=10^{9}\). In order to research the effect of \(\delta_{\mu}\) on the secret key rate for different \(N_{Z}\), we plot the \(R_{\delta_{\mu}}/R_{ideal}\) curves. The \(R_{ideal}\) means the result without intensity fluctuations and the \(R_{\delta_{\mu}}\) is the secure key rate under certain intensity fluctuations with the parameter \(\delta_{\mu}\). From the discussion above, it is obvious that the intensity fluctuation parameter \(\delta_{\mu}\) has a negligible impact on the secure key rate. The relations between \(R_{\delta_{\mu}}/R_{ideal}\) and the global attenuation with \(\delta_{\mu}\)= 0.01, 0.05, 0.09 for \(N_{Z}=10^{8}\) and \(N_{Z}=10^{9}\) are shown in Figure 3.  
Figure 1: (Color online) Secret key rate vs. global attenuation for \(N_{Z}=10^{8},10^{9},10^{10},10^{20}\) with \(\delta_{\mu}=0\), 0.09.