# Conditional Measurements with Silicon Photomultipliers
## 4 Discussion  
In order to validate the model for conditioning addressed in the previous Section, hereafter we present and discuss the experimental generation of nonclassical conditional states. As already explained in [6; 17; 18], such states can be obtained in post-processing by selecting a certain number of photons in one TWB arm and reconstructing the modified distribution of photons in the other arm. In Section 3 we showed that the unconditioned state is formally described by a multi-thermal distribution, which reduces to Equation (18) when the light in one arm is characterized by a very large number of modes [32; 33; 34] and is detected by a SiPM characterized by an OCT probability \(\epsilon\
eq 0\). In Figure 7 we show the detected-event distributions having mean values \(\langle k\rangle=2.63\) (panel (a)), 2.66 (panel (b)), 1.43 (panel (c)), and 0.57 (panel (d)). The experimental data are shown as gray dots, while the theoretical fitting functions according to Equation (18) are presented as gray lines. To quantify the agreement between the experimental data and the theoretical expectations we evaluate the fidelity \(f=\sum_{m=0}^{\tilde{m}}\sqrt{p^{\text{th}}(k)P(k)}\), in which \(P^{\text{th}}(k)\) and \(P(k)\) are the theoretical and experimental distributions, respectively, and the sum extends up to the maximum number of detected events \(k\) above which both \(P^{\text{th}}(k)\) and \(P(k)\) become negligible. Figure 7: Detected-event distributions \(P(k)\) of the unconditioned state having mean value \(\langle k\rangle=2.63\) (panel (a)), 2.66 (panel (b)), 1.43 (panel (c)), and 0.57 (panel (d)). The experimental data are shown as gray dots, while the theoretical expectations are presented as gray lines. The fidelity values are: \(f=0.9999\) in all panels. From the fitting procedure, it is possible to obtain the value of the only fitting parameter, namely the OCT. In particular, we notice that for the four considered measurements, the OCT value is of the same order of magnitude and always less than 1%, thus proving that the cross-talk probability affecting this model of SiPM is really small, even if not completely negligible. We remark that the estimated values for the OCT probability in Figure 8 are smaller than those reported in the datasheet of our sensors [30], but consistent with the characterization that we have already provided for these SiPMs in [16]. In order to prove that the conditioning procedure changes the statistical properties of such states making them sub-Poissonian, we calculate the Fano factor of the conditional states obtained from each of the four considered unconditioned states. Indeed, as mentioned in Section 2.1, F \(<1\) is a sufficient condition for nonclassicality. In Figure 8 we show the experimental Fano factors shown as dots plus error bars, while the theoretical fitting functions according to the second line of Equations (31) are shown as dashed lines with the same color choice. For all the fitting functions we left \(\eta\) and \(\alpha\) as free fitting parameters, while we used the same values of \(\varepsilon\) obtained from the fitting of the marginal distributions. In particular, in all cases we obtained a balance factor \(\alpha\sim 0.99\) and a quantum efficiency \(\eta\sim 0.14\). As a general statement, we note that the data corresponding to the conditioning value \(k_{i}=1\) are larger than 1 for the largest mean values. Such a behavior is in agreement with the theoretical expectation expressed by the second line of Equations (31) and the plots in panel (a) of Figure 6. Moreover, we emphasize that for the smallest mean value the conditioning operation is applied up to \(k_{i}=3\) because the number of experimental data is not sufficient to reliably build the states corresponding to \(k_{i}>3\).