characterizes the minimal value of transmission that could be distinguished from zero transmission, and therefore quantify the target detection sensitivity. According to the parameter estimation theory [@ly2017tutorial], the maximal likelihood estimation $\hat{\eta}_{p,MLE}$ could achieve the minimal estimation variance, which is given by the inverse of the corresponding Fisher information FI: $$\begin{gathered}
\hat{\eta}_{p,MLE} = \operatorname*{argmax}\limits_{0\le \eta_p\le 1} p_{\eta_p}(x) \\
\text{FI} = \sum\limits_{x'} p_{\eta_p}(x') ({\frac{d}{d\eta_p}}\text{log}p_{\eta_p}(x'))^2\end{gathered}$$ where $p_{\eta_p}(x')$ is the probability distribution function of the different experiment outcomes $x'$ and $x$ are the actual experiment outcome. The value of the Fisher information FI depends on many factors such as the environment loss, target object reflectivity, and the detection efficiency.\
The transmission estimation formalism could also be used for receiver operation characteristics (ROC) analysis, which is a widely adopted figure of merit for target detection. When the target object is present ($\mathbf{H_1}$) or absent ($\mathbf{H_0}$), the probe transmission estimation signal $\hat{\eta}_{p,MLE}$ could be modeled as: $$\begin{aligned}
\mathbf{H}_1&: \hat{\eta}_{p,MLE} =\eta_p+\mathbf{n}(\eta_p)\\ \nonumber
\mathbf{H}_0&: \hat{\eta}_{p,MLE} =\mathbf{n}(0)\end{aligned}$$ where $\mathbf{n}(\eta_p)$ is the estimation uncertainty as a function of the probe photon transmission $\eta_p$ ($\eta_p=0$ when the target object is absent). The presence or absence of the target object could be decided with certain detection probability $P_\text{d}$ and false alarm 