# Finite-Key Analysis of 1-Decoy Method Quantum Key Distribution with Intensity Fluctuation
## 2 The Finite-Key Analysis of 1-Decoy with Intensity Fluctuations
One can obtain the following relation:  
\[\mid M_{\alpha,\mu_{i}}-M_{\alpha,\mu_{i}}^{*}\mid\leq\delta_{A}(M_{\alpha}, \varepsilon_{A2}), \tag{6}\]  
with the probability at least \(1-2\varepsilon_{A2}\). Moreover, \(M_{\alpha}=\sum_{\mu_{i}\in\{\mu_{1},\mu_{2}\}}M_{\alpha,\mu_{i}}\) is the total number of errors in which [PERSON] sent states on \(\alpha\) basis. Thence, we can bound \(N_{\alpha,\mu_{i}}^{*}\) and \(M_{\alpha,\mu_{i}}\) for a given intensity \(\mu_{i}\in\{\mu_{1},\mu_{2}\}\), respectively, as follows:  
\[N_{\alpha,\mu_{i}}^{*L}=N_{\alpha,\mu_{i}}-\delta_{A}(N_{\alpha},\varepsilon_ {A1})\leq N_{\alpha,\mu_{i}}^{*}\leq N_{\alpha,\mu_{i}}+\delta_{A}(N_{\alpha},\varepsilon_{A1})=N_{\alpha,\mu_{i}}^{*L}, \tag{7}\]  
\[M_{\alpha,\mu_{i}}^{*L}=M_{\alpha,\mu_{i}}-\delta_{A}(M_{\alpha},\varepsilon_ {A2})\leq M_{\alpha,\mu_{i}}^{*}\leq M_{\alpha,\mu_{i}}+\delta_{A}(M_{\alpha},\varepsilon_{A2})=M_{\alpha,\mu_{i}}^{*L}. \tag{8}\]  
After error correction and privacy amplification, the upper bound on the secret key length of the 1-decoy can be derived as [21]:  
\[l\geq s_{Z,0}^{L}+s_{Z,1}^{L}(1-h(\phi_{Z}^{L}))-\lambda_{EC}-6 log(19/ \varepsilon_{sec})-log(2/\varepsilon_{cor}). \tag{9}\]  
where \(\lambda_{EC}\) is the number of bits leaked during the error correction step; \(\varepsilon_{sec}\) and \(\varepsilon_{cor}\) are the secrecy and correctness parameters. According to Equations (1) and (4), we can obtain the detailed expression of the boundary of \(N_{\alpha,\mu_{i}}^{*}\)  
\[p_{\mu_{i}}\sum_{n=0}^{\infty}\frac{e^{-\mu_{i}^{+}}\left(\mu_{i}^{-}\right)^{n }}{\tau_{n}^{+}n!}s_{\alpha,n}\leq N_{\alpha,\mu_{i}}^{*}\leq p_{\mu_{i}}\sum_{ n=0}^{\infty}\frac{e^{-\mu_{i}^{-}}\left(\mu_{i}^{+}\right)^{n}}{\tau_{n}^{-}n!}s_{ \alpha,n}. \tag{10}\]  
Thence, we can obtain the following bounds:  
\[\tau_{n}^{-}=\sum_{\mu_{i}\in\{\mu_{1},\mu_{2}\}}p_{\mu_{i}}e^{-{u_{i}^{+}}}( {u_{i}^{-}})^{n}/n!\leq\tau_{n}\leq\sum_{\mu_{i}\in\{\mu_{1},\mu_{2}\}}p_{\mu_ {i}}e^{-{u_{i}^{-}}}({u_{i}^{+}})^{n}/n!=\tau_{n}^{+}. \tag{2}\]0\[\tau_{n}^{-}=\sum_{\mu_{i}\in\{\mu_{1},\mu_{2}\}}p_{\mu_{i}}e^{-{u_{i}^{+}}}( {u_{i}^{-}})^{n}/n!\leq\tau_{n}\leq\sum_{\mu_{i}\in\{\mu_{1},\mu_{2}\}}p_{\mu_ {i}}e^{-{u_{i}^{-}}}({u_{i}^{+}})^{n}/n!=\tau_{n}^{+}. \tag{2}\]1  
By adding Equations (11) and (12), we can obtain  
\[\tau_{n}^{-}=\sum_{\mu_{i}\in\{\mu_{1},\mu_{2}\}}p_{\mu_{i}}e^{-{u_{i}^{+}}}( {u_{i}^{-}})^{n}/n!\leq\tau_{n}\leq\sum_{\mu_{i}\in\{\mu_{1},\mu_{2}\}}p_{\mu_ {i}}e^{-{u_{i}^{-}}}({u_{i}^{+}})^{n}/n!=\tau_{n}^{+}. \tag{2}\]2  
Even if the \(\mu_{1}^{-}\) is the lower of the \(\mu_{1}\) and the \(\mu_{2}^{+}\) is the upper of the \(\mu_{2}\), the relationship \(\mu_{1}^{-}>\mu_{2}^{+}\) still holds. Therefore, we can obtain  
\[\tau_{n}^{-}=\sum_{\mu_{i}\in\{\mu_{1},\mu_{2}\}}p_{\mu_{i}}e^{-{u_{i}^{+}}}( {u_{i}^{-}})^{n}/n!\leq\tau_{n}\leq\sum_{\mu_{i}\in\{\mu_{1},\mu_{2}\}}p_{\mu_ {i}}e^{-{u_{i}^{-}}}({u_{i}^{+}})^{n}/n!=\tau_{n}^{+}. \tag{2}\]3  
Therefore, we can obtain  
\[ \begin{split}\frac{e^{\mu_{1}^{+}}N_{Z,\mu_{1}}^{*}}{p_{\mu_{1}} }-\frac{e^{\mu_{2}^{-}}N_{Z,\mu_{2}}^{*}}{p_{\mu_{2}}}&\geq \frac{(\left(\mu_{1}^{-}\right)^{2}-\left(\mu_{2}^{+}\right)^{2})}{(\mu_{2}^{+ })^{2}}(\frac{e^{\mu_{2}^{+}}N_{Z,\mu_{2}}^{*}}{p_{\mu_{2}}}-\frac{s_{Z,0}}{ \tau_{0}^{-}})\\ &+(\frac{\left(\mu_{1}^{-}-\mu_{2}^{+}\right)}{\tau_{1}^{+}}- \frac{\left(\mu_{1}^{-}\right)^{2}-\left(\mu_{2}^{+}\right)^{2}}{\mu_{2}^{+} \tau_{1}^{-}})s_{Z,1}.\end{split}  \tag{15}\]  
It is easy to isolate \(s_{Z,1}\) from Equation (15):  
\[ \begin{split} s_{Z,1}\geq s_{Z,1}^{L}& =\frac{\tau_{1}^{+}\tau_{1}^{-}\mu_{2}^{+}}{\mu_{1}^{-}(\mu_{2}^{+} \tau_{1}^{-}-\mu_{1}^{-}\tau_{1}^{+})+(\mu_{2}^{+})^{2}(\tau_{1}^{+}-\tau_{1}^ {-})}(\frac{e^{\mu_{1}^{+}}N_{Z,\mu_{1}}^{*}}{p_{\mu_{1}}}-\frac{e^{\mu_{2}^{- }}N_{Z,\mu_{2}}^{*}}{p_{\mu_{2}}}\\ &\qquad\qquad\qquad\qquad-\frac{(\left(\mu_{1}^{-}\right)^{2}- \left(\mu_{2}^{+}\right)^{2})}{\left(\mu_{2}^{+}\right)^{2}}(\frac{e^{\mu_{2}^ {+}}N_{Z,\mu_{2}}^{*}}{p_{\mu_{2}}}-\frac{s_{Z,0}}{\tau_{0}^{-}})).\end{split}