post data processing. For simplicity of the following analysis, the detectors are assumed to have a perfect temporal resolution. To model the probability distribution of the different detection outcomes, it is convenient to consider a long quasi-CW pump pulse, during which the probability of generating a photon pair is $\bar{n}$. When a photon pair is generated, then the conditional joint probability distribution of detection time $t_p,t_r$ of the probe and reference photon, respectively, could be approximated as a bivariate Gaussian distribution. Then the joint probability $p_T$ of the detection outcomes is: $$\begin{gathered}
p_T(\text{no pairs}) = 1-\bar{n}\\
p_T(t_p,t_r) = \frac{\bar{n}}{2\pi\sigma_+\sigma_-} \exp(-\frac{(t_p-t_r)^2}{4\sigma^2_-}-\frac{(t_p+t_r)^2}{4\sigma^2_+})\end{gathered}$$ where $\sigma_+$ characterizes the duration of the quasi-CW pump pulse and $\sigma_-=\Delta t/\sqrt{2}$ quantifies the intrinsic temporal correlation time of the photon pairs. The normalized mutual information $\text{MI}_T$ corresponds to the probability distribution of the measurement outcomes is given by: $$\begin{gathered}
\text{MI}_\text{T} = \text{MI}_\text{I} -\frac{1}{2}{\text{log}_2}(1-(\frac{\sigma_+^2-\sigma_-^2}{\sigma_+^2+\sigma_-^2})^2)\end{gathered}$$ The additional term of $\text{MI}_\text{T}$ is the temporal correlation enhancement, which is inversely related to the pump duration $\sigma_+$ and directly related to the intrinsic temporal correlation time $\Delta t$. Ideally, to achieve large correlation enhancement, the intrinsic temporal correlation time $\Delta t$ should be as short as possible. However, it should be noted that 