## Appendix A The Bounds Estimation of \(P_{i}^{\alpha}\)  
The attenuation coefficient \(\eta_{\alpha}^{m}\) represents the total attenuation from the laser source to the output when the original signal contains m photons. Here, \(\alpha=s,d\) denotes the signal and decoy states, respectively. \(\eta_{\alpha}^{m}\) is given by the following equation  
\[\eta_{\alpha}=\frac{\mu_{\alpha}}{N}=\frac{2\mu_{\alpha}}{N_{\min}+N_{\max}}. \tag{10}\]  
The upper and lower bounds of \(P_{i}^{\alpha}(i=0,1,2)\) can be estimated theoretically. The elimination method is similar to the decoy scheme and can be used to scale the decay probability when \(N\in[N_{\min},N_{\max}]\). Firstly, \(P_{i}^{\alpha}(i=0,1,2)\) can be rewritten as  
\[P_{i}^{\alpha}=\sum_{N=0}^{N_{\min}-1}P(N)P_{N}^{\alpha}(i)+\sum_{N=N_{\min}}^ {N_{\max}}P(N)P_{N}^{\alpha}(i)+\sum_{N=N_{\max}+1}^{\infty}P(N)P_{N}^{\alpha} (i). \tag{11}\]  
Analyzing the upper and lower bounds of each component in Equation (11)  
\[\frac{C_{N+1}^{i}(1-\eta_{\alpha})^{N+1}}{C_{N}^{i}(1-\eta_{\alpha})^{N}}= \frac{N+1}{N+1-i}(1-\eta_{\alpha})=\left(1+\frac{i}{N+1-i}\right)(1-\eta_{ \alpha}). \tag{12}\]  
When i = 0,  
\[\frac{C_{N+1}^{0}(1-\eta_{\alpha})^{N+1}}{C_{N}^{0}(1-\eta_{\alpha})^{N}}<1, \tag{13}\]  
\[P_{N_{\max}}^{\alpha}\left(0\right)\leq P_{N}^{\alpha}\left(0\right)\leq P_{ N_{\min}}^{\alpha}\left(0\right). \tag{14}\]  
By substituting Equation (14) into Equation (24), we obtain  
\[P_{0}^{\alpha}\geq\sum_{N=N_{\min}}^{N_{\max}}P(N)P_{N}^{\alpha}(0)\geq\sum_ {N=N_{\min}}^{N_{\max}}P(N)C_{N_{\max}}^{0}\eta_{\alpha}^{0}(1-\eta_{\alpha}) ^{N_{\max}-0}, \tag{15}\]  
\[ \begin{split} P_{0}^{\alpha}&\leq\sum_{N=0}^{N_{ \min}-1}P(N)+\sum_{N=N_{\min}}^{N_{\max}}P(N)C_{N_{\min}}^{0}\eta_{\alpha}^{0} (1-\eta_{\alpha})^{N_{\min}-0}\\ &+\sum_{N=0}^{N_{\min}-1}P(N)C_{N_{\max}+1}^{0}\eta_{\alpha}^{0} (1-\eta_{\alpha})^{N_{\max}+1},\end{split}  \tag{16}\]  
because  
\[\sum_{N=N_{\min}}^{N_{\max}}P(N)C_{N_{\max}}^{0}\eta_{\alpha}^{0}(1-\eta_{ \alpha})^{N_{\max}-0}=\sum_{N=N_{\min}}^{N_{\max}}P(N)(1-\eta_{\alpha})^{N_{ \max}}\geq(1-\Delta)(1-\eta_{\alpha})^{N_{\max}}, \tag{17}\]  
\[ \begin{split}&\sum_{N=0}^{N_{\min}-1}P(N)+\sum_{N=N_{\min}}^{N_{ \max}}P(N)C_{N_{\min}}^{0}\eta_{\alpha}^{0}(1-\eta_{\alpha})^{N_{\min}-0}+ \sum_{N=0}^{N_{\min}-1}P(N)C_{N_{\max}+1}^{0}\eta_{\alpha}^{0}(1-\eta_{\alpha })^{N_{\max}+1}\\ &\leq\Delta+(1-\Delta)(1-\eta_{\alpha})^{N_{\min}}+\Delta(1-\eta _{\alpha})^{N_{\max}+1},\end{split}  \tag{18}\]  
by substituting Equation (17) and Equation (18) into Equation (15) and Equation (16), respectively, we can obtain the upper and lower bounds of \(P_{0}^{\alpha}\):  
\[P_{0}^{\alpha}\geq(1-\Delta)(1-\eta_{\alpha})^{N_{\max}}, \tag{19}\]  
\[P_{0}^{\alpha}\leq\Delta+(1-\Delta)(1-\eta_{\alpha})^{N_{\min}}+\Delta(1-\eta _{\alpha})^{N_{\max}+1}. \tag{20}\]Likewise, for Equation (A3), when i = 1,  
\[\frac{C_{N+1}^{1}(1-\eta_{\alpha})^{N+1}}{C_{N}^{1}(1-\eta_{\alpha})^{N}}=\frac{N +1}{N}\bigg{(}1-\frac{2\mu_{\alpha}}{N_{\text{min}}+N_{\text{max}}}\bigg{)},\] (A12)  
it can be seen that Equation (A12) decreases with \(N\). Here, we make it equal to 1,  
\[\frac{N+1}{N}\bigg{(}1-\frac{2\mu_{\alpha}}{N_{\text{min}}+N_{\text{max}}} \bigg{)}=1,\] (A13)  
we can determine the inflection point \(N\) of Equation (A12)  
\[N=\widetilde{N}=\frac{N_{\text{min}}+N_{\text{max}}}{2\mu_{\alpha}}-1.\] (A14)  
In a QKD system, it is typically set that \(\mu_{\alpha}<1\), which implies that \(\widetilde{N}>N_{\text{max}}\). Based on the above result, the upper and lower bounds of \(P_{1}^{\alpha}\) can be calculated.