# Conditional Measurements with Silicon Photomultipliers
## 3 Results
### The Effects of the OCT on the Photon-Number Statistics of the Conditional State  
Here we investigate the effects of the OCT on the statistics of the signal after conditioning over the idler, as described in Section 2.1. We also consider the effect of the imbalance between the quantum efficiencies of the signal and idler detectors. Note that, while we remove the assumption that the detectors of the two parties share the same quantum efficiency \(\eta=\eta_{s}=\eta_{i}\), we keep assuming the same OCT probability \(\varepsilon=\varepsilon_{s}=\varepsilon_{i}\). The imbalance is introduced through the parameter \(\alpha\equiv\eta_{s}/\eta_{i}\), with \(\eta\equiv\eta_{i}\). The expression of the reduced state of the signal after measuring \(k_{i}\) events over the idler is straightforward from Equation (11) and reads  
\[ \begin{split}\hat{\rho}_{s}^{(k_{i})}=&\frac{(1- \lambda^{2})^{\mu}}{\mathrm{p}(\mathrm{k_{i}})}\bigg{(}\frac{\varepsilon}{1- \varepsilon}\bigg{)}^{k_{i}}\sum_{m_{i}=[\frac{k_{i}}{2}]}^{k_{i}}\binom{m_{i} }{k_{i}-m_{i}}\bigg{(}\frac{(1-\varepsilon)^{2}}{\varepsilon}\frac{\eta}{1- \eta}\bigg{)}^{m_{i}}\\ &\bigotimes_{j=1}^{\mu}\sum_{l_{j}}\delta_{m_{i},\gamma}\sum_{n_ {j}=l_{j}}^{\infty}\binom{n_{j}}{m_{j}}\lambda^{2n_{j}}(1-\eta)^{n_{j}}|n_{j} \rangle\langle n_{j}|\end{split}  \tag{20}\]  
where again \(\gamma\equiv\sum_{j=1}^{\mu}l_{j}\) and \([\cdot]\) is the ceiling function. Note that the conditional state correctly does not depend on \(\alpha\) since no detection over the signal party has occurred yet. On the contrary, the related detected-event distribution is a function of \(\alpha\), other than of the number of events detected over the idler \(k_{i}\):  
\[ \begin{split}\mathrm{p}^{(k_{i})}(k_{s})=&\frac{(1- \lambda^{2})^{\mu}}{\mathrm{p}(\mathrm{k_{i}})}\bigg{(}\frac{\varepsilon}{1- \varepsilon}\bigg{)}^{k_{i}+k_{s}}\sum_{m_{i}=[\frac{k_{i}}{2}]}^{k_{i}}\sum_{m _{s}=[\frac{k_{s}}{2}]}^{k_{s}}\binom{m_{i}}{k_{i}-m_{i}}\bigg{(}\frac{m_{s}}{ \varepsilon}\frac{(1-\varepsilon)^{2}}{1-\eta}\bigg{)}^{m_{i}}\\ &\bigg{(}\frac{(1-\varepsilon)^{2}}{\varepsilon}\frac{\alpha\eta }{1-\alpha\eta}\bigg{)}^{m_{s}}\sum_{l=m_{i}}^{\infty}\binom{l+\mu-1}{l}\binom {l}{m_{s}}\binom{l}{m_{i}}\lambda^{2l}(1-\eta)^{l}(1-\alpha\eta)^{l}.\end{split}  \tag{21}\]  
In the limit of large number of modes, we find  
\[ \begin{split}\mathrm{p}^{(k_{i})}_{\mu\to\infty}(k_{s})=& \frac{\exp\Bigl{(}-\frac{\langle k_{i}\rangle}{\eta(1+\varepsilon)} \Bigr{)}}{\mathrm{p}_{\mu\to\infty}(k_{i})}\bigg{(}\frac{\varepsilon}{1- \varepsilon}\bigg{)}^{k_{i}+k_{s}}\sum_{m_{i}=[\frac{k_{i}}{2}]}^{k_{i}}\sum_{m _{s}=[\frac{k_{s}}{2}]}^{k_{s}}\binom{m_{i}}{k_{i}-m_{i}}\binom{m_{s}}{k_{s}-m _{s}}\\ &\bigg{(}\frac{(1-\varepsilon)^{2}}{\varepsilon}\frac{\eta}{1- \eta}\bigg{)}^{m_{i}}\bigg{(}\frac{(1-\varepsilon)^{2}}{\varepsilon}\frac{ \alpha\eta}{1-\alpha\eta}\bigg{)}^{m_{s}}\sum_{l=m_{s}}^{\infty}\frac{1}{l!} \binom{l}{m_{s}}\binom{l}{m_{i}}\\ &\bigg{(}\frac{\langle k_{i}\rangle(1-\eta)(1-\alpha\eta)}{\eta( 1+\varepsilon)}\bigg{)}^{l}.\end{split}  \tag{22}\]  
Given Equation (21), we can have access to every moment of the conditional-state distribution. For instance, the first moment reads  
\[\langle\hat{k}_{s}\rangle^{(k_{i})}=\frac{\alpha\eta(1+\varepsilon)}{1-\lambda ^{2}(1-\eta)}\bigg{[}k_{i}+\mu\lambda^{2}(1-\eta)-\frac{\partial}{\partial x} \log\chi(x)\bigg{|}_{x=0}\bigg{]} \tag{23}\]  
where  
\[\chi(x)\equiv\sum_{l=0}^{\lfloor\frac{k_{i}}{2}\rfloor}\binom{k_{i}+\mu-1-l}{ \mu-1}\binom{k_{i}-l}{l}\bigg{(}\frac{1-\lambda^{2}(1-\eta)}{(1-\varepsilon)^{ 2}\eta\lambda^{2}}\varepsilon\bigg{)}^{l}e^{lx} \tag{24}\]  
is a sort of characteristic function related to the discrete probability distribution in Equation (17).