# Conditional Measurements with Silicon Photomultipliers
## 3 Results
### The Effects of the OCT on the Nonclassicality of the Conditional State  
Finally, we focus on the nonclassicality of the state generated after post-selection and evaluate to what extent the OCT is detrimental for this quantum resource. The first and the second moments of the conditional-state distribution allow us to retrieve the Fano factor for the detected events by means of Equation (14) expressed for the operator \(\hat{k}_{s}\). As mentioned in Section 2.1, the Fano factor provides a sufficient condition for nonclassicality. For the distribution of the conditional state in Equation (21) we find that it reads  
\[ \begin{split}\mathrm{F}_{s}^{(k)}=&\frac{1+3 \epsilon}{1+\epsilon}-\alpha\eta(1+\epsilon)+\frac{1}{\langle\hat{k}_{s} \rangle^{(k_{i})}}\bigg{[}\frac{\alpha\eta(1+\epsilon)}{1-\lambda^{2}(1-\eta )}\bigg{]}^{2}\Big{[}\lambda^{2}(1-\eta)(k_{i}+\mu)\Big{]}-\\ &\frac{1}{\langle\hat{k}_{s}\rangle^{(k_{i})}}\bigg{[}\frac{ \alpha\eta(1+\epsilon)}{1-\lambda^{2}(1-\eta)}\bigg{]}^{2}\cdot\frac{\partial }{\partial x}\log\chi(x)\bigg{[}\lambda^{2}(1-\eta)-\frac{\partial}{\partial x }\log\Big{(}\frac{\partial}{\partial x}\log\chi(x)\Big{)}\bigg{]}\bigg{|}_{x= 0}\end{split}  \tag{27}\]  
where \(\chi(x)\) is defined in Equation (24). Again, we highlight that Equation (27) can be written as a function of experimental quantities by just replacing \(\lambda\) with \(\langle k_{i}\rangle\) through Equation (10). Since the expression is quite complex, in Figure 6 we show the behavior of \(\mathrm{F}_{s}^{(k_{i})}\) as a function of the conditioning value \(k_{i}\) for different choices of the other parameters: in panel (a), different mean values of the unconditioned state \(\langle k_{i}\rangle\), in panel (b), different values of the balance parameter \(\alpha\), in panel (c), different choices of the cross-talk probability \(\epsilon\), and finally in panel (d), different number of modes of the unconditioned state \(\mu\). It is worth noting that the subPoissonianity of the Fano factor can be increased by decreasing the mean value of the unconditioned state and increasing the number of modes, and by operating on the features of the detectors, namely reducing the OCT probability and increasing the balance factor. Again, if \(\epsilon=0\) and \(\alpha=1\), we retrieve the known expression of the Fano factor for the conditional state in the context of multi-mode TWB states and limited quantum efficiency, as outlined in [6], i.e.,  
\[\mathrm{F}_{s}^{(k_{i})}(\epsilon=0,\alpha=1)=(1-\eta)\bigg{[}1+\frac{\langle \hat{k}_{i}\rangle(k_{i}+\mu)(\langle\hat{k}_{i}\rangle+\eta\mu)}{(\langle\hat {k}_{i}\rangle+\mu)[(k_{i}+\mu)(\langle\hat{k}_{i}\rangle+\eta\mu)-\eta\mu( \langle\hat{k}_{i}\rangle+\mu)]}\bigg{]}. \tag{28}\]  
Note that Equation (27) can be significantly simplified by taking the limit to realistic values for the parameters \(\mu\) and \(\epsilon\). As mentioned above, our experimental conditions allow us to take the limit \(\mu\rightarrow\infty\), which reduces the sum in Equation (24) to  
\[\chi(0)\sim\bigg{(}\langle k_{i}\rangle\frac{1-\epsilon}{1+\epsilon}\bigg{)}^ {k_{i}}\sum_{l=0}^{\lfloor\frac{k_{i}}{2}\rfloor}\frac{1}{l!(k_{i}-2l)!}\bigg{(} \frac{\epsilon(1+\epsilon)}{\langle\hat{k}_{i}\rangle(1-\epsilon)^{2}}\bigg{)} ^{l}, \tag{29}\]  
but then, being the typical OCT probabilities of modern SiPMs of the order \(10^{-2}\), the largest order in the argument of the sum for a given term \(l\) is  
\[\frac{(\epsilon/\langle\hat{k}_{i}\rangle)^{l}}{l!(k_{i}-2l)!}. \tag{30}\]Thus, provided that the order of the mean number of detected events is larger than the order of \(\varepsilon\), the argument of the sum gets smaller as \(l\) increases. If we keep the \(l=0\) term only, all the logarithmic derivatives of \(\chi(x)\) are null, so that the mean value and the Fano factor of the conditional state are much simplified.