# Sending-or-Not-Sending Twin-Field Quantum Key Distribution with Light Source Monitoring
## 3 Performance with Numerical Simulation  
The performance of the original SNS protocol and the SNS protocol with the LSM scheme can be compared with a determined light source condition. Considering an ideal simulation circumstance  
Figure 1: The structure of the sending-or-not-sending (SNS) protocol with an extra LSM module in each of [PERSON]’s and Bob’s parts. The LSM module is made up of a variable optical attenuators (VOA) and a single photon detector (SPD). By changing the attenuation coefficient of the VOA, various sets of the results on the responding probability of the SPD are obtained, which can be used to estimate \(P_{n}(\mu)\) effectively. The details of the monitoring scheme have been discussed in Reference [38]. first [22], the PND of the light source, \(P_{n}(\mu)\), is assumed to satisfy Equation (12). In this case, the probabilities \(P^{\mu}(\eta)\) in the LSM scheme can be simulated as [38]  
\[P^{\mu}(\eta)=(1-Y_{0})e^{-\eta\mu}, \tag{23}\]  
and the estimation results of \(P_{n}(\mu)\) can be calculated with Equations (17)-(21). Except \(P^{\mu}(\eta)\), other parameters used in Equations (9)-(11), (14) and (15), such as \(S_{\mu_{d_{\mathrm{g}}}},E_{\mu_{d_{\mathrm{g}}}},S_{Z},E_{Z}\), can be simulated with the same method discussed in the original SNS protocol [22; 27], hence the final secret key rate can be calculated. To compare the performance of the LSM scheme and the original SNS protocol under ideal simulation conditions, the simulation parameters, which are listed in Table 1, are set to be the same as in Reference [22]. Besides, for the LSM scheme, we set the attenuation coefficient as \(\eta_{0}=1\), \(\eta_{1}=0.95\), \(\eta_{2}=0.9\) to obtain tight estimation results of \(P_{n}(\mu)\); other parameters, including the intensities \(\mu_{s},\mu_{d_{1}},\mu_{d_{2}}\) and the sending probability \(\epsilon\), are optimized at different transmission distances for both the original protocol and the SNS protocol with the LSM scheme. The simulation results shown in Figure 2 indicate that, with the LSM scheme, the SNS protocol under UPC can have a performance that is almost the same as that of the original protocol with an ideal source. Specifically, both cases have a maximum transmission distance of up to over 800 km, and the difference between the LSM scheme and the ideal source condition is only about 0.5 km. Besides, the difference of the secret key rate between them is only about 1% for a typical distance of 100 km. In order to further analyze the practical performance of the SNS protocol and it with the LSM scheme, it is necessary to consider a more practical simulation circumstance. For a real-life QKD system, the PND of the source's signals is not fixed although without [PERSON]'s disturbance [31; 40; 41]. Under this condition, the performance of the original SNS protocol will be degraded even though the security problem of an unknown PND is ignored. Specifically, the signal sent out from the source can be considered as a fluctuated coherent state with a Gaussian-distributed average photon number \(\mu\), which has a probability distribution of \begin{table}
\begin{tabular}{l c c c c} \hline \hline \(\alpha\) & \(Y_{0}\) & \(\eta_{D}\) & \(e_{\mathrm{det}}\) & \(f\) \\ \hline
0.2 & \(1.0\times 10^{-11}\) & 80\% & 1\% & 1.1 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Values of parameters used in simulation. \(\alpha\): the fiber loss coefficient (unit: dB/km); \(Y_{0}\): the dark count rate of the detector; \(\eta_{D}\): the detection efficiency; \(e_{\mathrm{det}}\): the misalignment error of the QKD system; \(f\): the error correction efficiency. Figure 2: The performance of the proposed LSM scheme (red dash curve) compared to original SNS protocol (blue dash curve) with the parameters set as Table 1. The ratios of secret key rate between the LSM scheme and original SNS protocol are about 99.1%, 98.9%, 98.8% at the distance of \(100,300,500\) km, and the maximum transmission distances of the LSM scheme and original SNS protocol are about 806.0 and 806.5 km.