# Conditional Measurements with Silicon Photomultipliers
## 3 Results
### The Effects of the OCT on the Nonclassicality of the Conditional State
By taking this limit, we neglect the OCT contribution provided by the asymmetry between odd and even detected events, which is reasonable if \(\varepsilon\) is small (i.e., \(\varepsilon<0.1\)), as highlighted in Figures 4 and 5. Given this approximation and the limit for \(\mu\), one gets  
\[ \begin{split}&\langle\dot{k}_{s}\rangle_{\mu\to\infty}^{(k_{i})} =\alpha[\eta(1+\varepsilon)k_{i}+(1-\eta)\langle\dot{k}_{i}\rangle]\\ &\mathrm{F}_{\varepsilon_{\mu\to\infty}}^{(k_{i})}=\frac{1+3 \varepsilon}{1+\varepsilon}-\alpha\eta(1+\varepsilon)\Bigg{[}1-\frac{(1- \eta)\langle\dot{k}_{i}\rangle}{\eta(1+\varepsilon)k_{i}+(1-\eta)\langle \dot{k}_{i}\rangle}\Bigg{]}.\end{split}  \tag{31}\]Equations (31) allow us to find the threshold conditioning value \(\bar{k}\) such that the detected state is nonclassical, i.e., \(\mathrm{F}^{(k_{i}<\bar{k})}_{s_{p\to\infty}}<1\). Before retrieving \(\bar{k}\), we remark that in the limit \(\mu\to\infty\), the Fano factor in Equation (28), where \(\varepsilon=0\) and \(\alpha=1\), is a function of the quantum efficiency only, i.e.,  
\[\mathrm{F}^{(k_{i})}_{s_{p\to\infty}}(\varepsilon=0,\alpha=1)=\frac{1-\eta}{ 1-\eta/2}. \tag{32}\]  
However, \(0\leq\mathrm{F}^{(k_{i})}_{s_{p\to\infty}}(\varepsilon=0,\alpha=1)\leq 1\ \forall\eta\in[0,1]\), which means that the imperfections in detection due to limited quantum efficiency never provide a detected superPoissonian statistics in this context. Only in the limit case \(\eta=0\) the nonclassicality of a conditional state from multi-mode TWB is not revealed by the Fano factor, otherwise the detected nonclassicality is just reduced with respect to the ideal case (\(\eta=1\)). On the contrary, the OCT can completely conceal the quantum nature of a conditional state since we may have \(\mathrm{F}^{(k)}_{s_{p\to\infty}}>1\) for some \(k_{i}<\bar{k}\) where  
\[\bar{k}=\frac{2\varepsilon(1-\eta)\langle\hat{k}_{i}\rangle}{\eta(1+ \varepsilon)[\alpha\eta(1+\varepsilon)^{2}-2\varepsilon]}. \tag{33}\]  
Note that \(\bar{k}>0\iff\alpha\eta(1+\varepsilon)^{2}>2\varepsilon\), i.e., if  
\[\eta_{s}>\eta_{th}(\varepsilon)\equiv\frac{2\varepsilon}{(1+\varepsilon)^{2}} \tag{34}\]  
where we replaced \(\alpha\eta\) with the quantum efficiency of the detector of the signal party \(\eta_{s}\) through the definition of \(\alpha\). Therefore, Equation (33) shows that for \(\varepsilon>0\) and \(\eta<1\) there is a conditioning number \(\bar{k}>0\) such that if \(k_{i}<\bar{k}\) the detected statistics is superPoissonian (see Figure 6). Moreover, Equation (34) gives an experimental condition for the observation of the nonclassicality of the conditional state: provided that \(\eta_{s}\) is larger than the threshold \(\eta_{th}\), then a finite \(\bar{k}\) exists such that one can measure \(\mathrm{F}^{(k_{i})}_{s_{p\to\infty}}<1\ \forall k_{i}>\bar{k}\). Note that \(\bar{k}(\varepsilon=0,\eta\
eq 0)=0\), which implies that the detected statistics is subPoissonian, if the only detection imperfection is a non-unit \(\eta>0\). However, it is remarkable that in the ideal case \(\eta=1\) we have a subPoissonian statistics independently of \(\varepsilon\), while if \(\eta\to 0\) and \(\varepsilon\
eq 0\), then the detected statistics is always superPoissonian, independently of \(k_{i}\). One may ask if a combination of \(\eta\) and \(\varepsilon\) exists such that \(\mathrm{F}^{(k_{i})}_{s_{p\to\infty}}=0\) for some \(k_{i}\). Unfortunately, this is not the case since in the second line of Equations (31) the Fano factor is a monotone decreasing function of \(k_{i}\) and it converges to an asymptotic value which is strictly positive \(\forall\varepsilon>0\). Finally, we remark that the threshold in Equation (34) is directly connected to the sub-Poissonianity of the original state, which in turn depends on its intrinsic nonclassical correlations. In fact the same threshold can be shown to hold for the observation of sub-shot-noise correlations of TWB.