# Finite-Key Analysis of 1-Decoy Method Quantum Key Distribution with Intensity Fluctuation
## 2 The Finite-Key Analysis of 1-Decoy with Intensity Fluctuations  
The 1-decoy method was presented as a simpler method with only two intensities, i.e., a signal and a decoy state, denoted as \(\mu_{1}\) and \(\mu_{2}\), respectively. In practical scenarios, the intensity fluctuations are usually caused by imperfections of the laser and the intensity modulators. Therefore, it is too difficult to use a specific analytical method to describe the intensities' fluctuations, but the range of the intensity can be assumed as \(\mu_{i}\in[{u_{i}}^{-},{u_{i}}^{+}],\mu_{i}\in\{\mu_{1},\mu_{2}\}\). Furthermore, we can characterize intensity fluctuations by \({u_{i}}^{-}=(1-\phi_{\mu})\mu_{i}\) and \({u_{i}}^{+}=(1+\delta_{\mu})\mu_{i}\), where the parameter \(\delta_{\mu}\) is used to quantify the degree of intensity fluctuation. Once the source is determined, the \(\delta_{\mu}\) can be regarded as constant. For each intensity, the probability that the light source produces the n-photon states is \(p_{n|\mu_{i}}\), which can be written as [18]  
\[\frac{e^{-{u_{i}}^{+}}({u_{i}}^{-})^{n}}{n!}\leq p_{n|\mu_{i}}=\frac{e^{-{u_{i }}}{u_{i}}^{n}}{n!}\leq\frac{e^{-{u_{i}}^{-}}({u_{i}}^{+})^{n}}{n!}. \tag{1}\]  
When the intensity fluctuations are not considered, the total probability that [PERSON] sends n-photon states can be expressed as \(\tau_{n}=\sum_{\mu_{i}\in\{\mu_{1},\mu_{2}\}}p_{\mu_{i}}e^{-{\mu_{i}}}{u_{i}}^ {n}/n!\). However, intensity fluctuations of modulation will affect the probability \(\tau_{n}\), and we can obtain the range of the \(\tau_{n}\) as follows:  
\[\tau_{n}^{-}=\sum_{\mu_{i}\in\{\mu_{1},\mu_{2}\}}p_{\mu_{i}}e^{-{u_{i}^{+}}}( {u_{i}^{-}})^{n}/n!\leq\tau_{n}\leq\sum_{\mu_{i}\in\{\mu_{1},\mu_{2}\}}p_{\mu_ {i}}e^{-{u_{i}^{-}}}({u_{i}^{+}})^{n}/n!=\tau_{n}^{+}. \tag{2}\]  
We set \(p_{\mu_{i}}\) to represent the probability of [PERSON] choosing intensity \(\mu_{i}\) and the probability \(p_{\mu_{i}|n}\) denotes the probability of n-photon coming from specific intensity \(\mu_{i}\) in the total n-photon detection events. \[\frac{p_{\mu_{i}}e^{-\mu_{i}^{+}}\left(\mu_{i}-\right)^{n}}{\tau_{n}^{+}n!}\leq p_{ \mu_{i}|n}=\frac{p_{\mu_{i}}e^{-\mu_{i}}\mu_{i}^{\,n}}{\tau_{n}n!}\leq\frac{p_{ \mu_{i}}e^{-\mu_{i}^{-}}\left(\mu_{i}^{+}\right)^{n}}{\tau_{n}^{-}n!}. \tag{3}\]  
Let \(N_{\alpha}\) be the total number observed by [PERSON] on \(\alpha\in\{Z,X\}\) basis. The \(s_{\alpha,n}\) are the total detections observed by [PERSON] in which [PERSON] sent n-photon states on \(\alpha\in\{Z,X\}\) basis. The number of detection events with a specific intensity \(\mu_{i}\) should be \(N_{\alpha,\mu_{i}}^{*}\) and the \(N_{\alpha,\mu_{i}}\) is the expectation of the amount of detection events. Then, we have  
\[N_{\alpha}=\sum_{\mu_{i}\in\{\mu_{1},\mu_{2}\}}N_{\alpha,\mu_{i}}=\sum_{n=0}^ {\infty}s_{\alpha,n}. \tag{4}\]  
In this paper, the effects of statistical fluctuations and intensity fluctuations are taken into account concurrently. We use the concentration inequality to describe the relationship between expected and observed values. Significantly, the intensity fluctuations will destroy the independence between response events. Therefore, we should use [PERSON]'s inequality [20], which holds with dependent random samples, to characterize the relation between the expected values and the observed values. The observed values \(N_{\alpha,\mu_{i}}^{*}\) shall be deviated from the expected values \(N_{\alpha,\mu_{i}}\) by  
\[\mid N_{\alpha,\mu_{i}}-N_{\alpha,\mu_{i}}^{*}\mid\leq\delta_{A}(N_{\alpha}, \varepsilon_{A1}), \tag{5}\]  
with the probability of at least \(1-2\varepsilon_{A1}\), where the \(\delta_{A}(x,y)=\sqrt{2 xln(\frac{1}{y})}\). Moreover, \(N_{\alpha}=\sum_{\mu_{i}\in\{\mu_{1},\mu_{2}\}}N_{\alpha,\mu_{i}}\) is the total number of events when [PERSON] and [PERSON] both choose \(\alpha\) basis. Simultaneously, \(M_{\alpha,\mu_{i}}^{*}\), the number of errors observed on \(\alpha\) basis with the intensity \(\mu_{i}\), can be estimated with the same method.