are the number of photon detection events $N_p,N_r$ on the probe and reference detector that do not contribute to coincidence detections. It could be shown that $N_c,N_p,N_r$ each obeys an independent Poisson distribution. The Fisher information for transmission estimation could be calculated from the joint probability distribution of $N_p,N_c$ and $N_r$ [@liu2019enhancing]: $$\begin{gathered}
\text{FI} = (\frac{\eta_r^2\nu^2}{P_c}+\frac{(1-\eta_r)^2\nu^2}{P_p}+\frac{\eta_r^2\nu^2}{P_r})\tau\\
P_c =  \eta_r\eta_p \nu+\eta_r\nu_b\nu T_c\\
P_r =  \eta_r \nu - P_c\hspace{20pt}\\
P_p = \nu_b+ \eta_p\nu - P_c \label{eq1}\end{gathered}$$ The definitions of different variables are summarized in table \[TAB\]. The Fisher information $\text{FI}$ increases as the coincidence window $T_c$ decreases, suggesting that the performance of the QTC protocol is related to the temporal correlation. The minimal coincidence window is limited by the intrinsic temporal correlation $\Delta t$ of the twin beams as well as the detector temporal resolution [@liu2019enhancing]. It worth noting that the CI protocol, which only consists of intensity detection on the probe detector, could be considered as a special case of the QTC protocol where the transmission of the reference photon $\eta_r$ is set to zero.

  $P_c$      coincidence detection rate (Hz)
  ---------- --------------------------------------------------
  $P_r$      reference detector detection rate (Hz)
  $P_p$      probe detector detection rate (Hz)
  $\nu$      probe photon generation rate (Hz)
  $\nu_b$    