## 2 SNS Protocol with UB
### Parameters Estimation with UB  
In the original SNS protocol, [PERSON] (Bob) sends a coherent signal [25], where the average number of photons follows a Poisson distribution  
\[P_{n,A}(\mu)=P_{n,B}(\mu)=P_{n}(\mu)=e^{-\mu}\frac{\mu^{n}}{n!}, \tag{21}\]  
we can then calculate Equations (5)-(7) using Equation (21). However, under untrusted light conditions, the photon distribution no longer follows a Poisson distribution. Therefore, light source monitoring is required to estimate \(p_{0}^{L(U)}(\mu_{i})\), \(p_{1}^{L(U)}(\mu_{i})\) and \(p_{2}^{L(U)}(\mu_{i})\). The light structure of the SNS protocol is similar to that of the MDI protocol with both protocols sharing similarities in phase and intensity modulation. The details of the SNS protocol, which is based on the UB monitoring scheme, are shown in Figure 1. The communicating parties estimate the probabilities of zero-, one-, and two-photon signals \(P_{k,A}(\mu),P_{k,B}(\mu)(k=0,1,2)\). Specifically, the monitoring parameter can be obtained through the UB light monitoring structure, which is the UB signal ratio, and it is denoted as \(\Delta\). \(\Delta\) is expressed by the following inequality:  
\[1-\Delta\leq\sum_{N=N_{\min}}^{N_{\max}}P(N)\leq 1. \tag{22}\]  
The UB signal is defined as photons \(n_{a}\) and \(n_{b}\), which are prepared by the communicating parties and located in the signal intervals \(\left[N_{\min}^{A},N_{\max}^{A}\right]\) and \(\left[N_{\min}^{B},N_{\max}^{B}\right]\), respectively. \(1-\Delta\) represents the probability that both communicating parties send UB signals. By  
Figure 1: The structure of the sending-or-not-sending (SNS) protocol with an extra UB module in each of [PERSON]’s and Bob’s parts. The UB module is composed of an attenuator (ATT) and a photon detector (PD). Other calculation parameters are similar to those in the original SNS [25; 26] protocol. To accurately estimate the performance of the UB light monitoring scheme, it is necessary to select an appropriate light source signal range based on experimental conditions to determine the UB signal proportion, which is followed by the calculation of the ideal secure bit rate. To simulate the experimental environment, we select the average photon number \(\bar{N}=1\times 10^{7}\), the light intensity fluctuation parameter \(\sigma_{0}=1\%\) and the interval parameter \(\delta=0.043\). Additionally, the traversal interval is set to \([N_{min},N_{max}]=[(1-\delta)N,(1+\delta)N]\) followed by performance simulation under these conditions. Based on the parameter estimation results from the previous section, simulations of the SNS protocol using the UB monitoring scheme can be performed under untrusted source conditions. First, we perform a performance simulation of the SNS protocol using ideal experimental parameters, which are set to be the same as in Reference [25] and listed in Table 1. The simulation results are shown in Figure 2. They demonstrate that the protocol can achieve performance close to that of an ideal light source environment, even under untrusted source conditions, when the UB light source monitoring is introduced. Under the set monitoring conditions, the maximum secure transmission distance can reach 817.5 km, which is over 95% of the ideal light source environment. Furthermore, we can analyze the performance of this monitoring scheme under untrusted source conditions and actual simulation parameters. The simulation results are shown in Figure 3. To simulate a real-world QKD system, we also consider factors such as a high dark count rate and low detection efficiency. Additionally, factors such as fiber attenuation, inherent system errors, and non-ideal data coordination efficiency are also considered in a non-ideal environment. Finally, the experimental parameters are chosen based on Table 2. The simulation result shows that under untrusted source conditions, the maximum secure transmission of the SNS protocol based on the UB monitoring scheme can reach 490 km, which is over 92% of the ideal light source environment.