# Conditional Measurements with Silicon Photomultipliers
## 4 Discussion
In order to explore in which way the conditional measurements modify the statistical properties of the unconditioned states in the presence of the OCT, in the two panels of Figure 9 we show some conditional distributions at different conditioning values together with the corresponding unconditioned statistics having mean values \(\langle k\rangle=2.66\) (panel (a)), and 1.43 (panel (b)). The data are presented as colored dots plus error bars (\(k_{i}=1\) in black, \(k_{i}=2\) in red, and \(k_{i}=3\) in blue), while the theoretical expectations  
Figure 8: Fano factor as a function of the conditioning value for four different unconditioned states having mean values \(\langle k\rangle=2.63\) (black), 2.66 (red), 1.43 (blue), and 0.57 (magenta). The experimental data are shown as dots plus error bars, while the theoretical fitting functions according to the second line of Equations (31) are presented as dashed curves with the same color choice. The fitting parameters are the following: \(\eta=0.134\), \(\alpha=0.990\) (black curve), \(\eta=0.157\), \(\alpha=0.989\) (red curve), \(\eta=0.158\), \(\alpha=0.997\) (blue curve), and \(\eta=0.125\), \(\alpha=0.986\) (magenta curve). The reduced \(\chi^{(2)}\) are: 0.34 (black curve), 0.14 (red curve), 0.94 (blue curve), and 0.05 (magenta curve). are shown as solid lines with the same color choice. The theoretical curves have been calculated according to Equation (22) using the parameter values of \(\varepsilon\) obtained from the fit of the unconditioned states (see caption of Figure 7) and those of \(\eta\) and \(\alpha\) obtained from the fit of the Fano (see caption of Figure 8). For the sake of clarity, in each panel of Figure 9 we show again the statistics of the unconditioned state as gray dots and the theoretical expectation as gray surface defined by dashed line. As expected from the two panels of the figure, it clearly appears that the conditional measurements change the statistics of the input state. Even in this case, to quantify the agreement between the experimental data and the theoretical expectations we evaluate the fidelity. We note that the higher the conditioning value the lower the fidelity value. This fact can be ascribed to the limited number of data at our disposal to build the statistics, which is lower and lower at increasing values of \(k_{i}\). Larger acquisitions of data could overcome such a limit. At the same time, the good dynamic range of SiPMs would suggest that both the unconditioned states and the corresponding conditional ones could be more populated, thus allowing us to really explore the mesoscopic intensity domain. Figure 9: Detected-event distributions \(P(k)\) of the conditional states for \(k_{i}=1\) (black curve), \(k_{i}=2\) (red curve), and \(k_{i}=3\) (blue curve) obtained from an unconditioned state having mean value \(\langle k\rangle=2.66\) (panel (**a**)) and \(1.43\) (panel (**b**)). The experimental data are shown as colored symbols, while the theoretical expectations are presented as solid lines with the same color choice. The fidelity values are: \(f=0.9999\), \(0.9961\), and \(0.9888\) in panel (**a**) and \(f=0.9999\), \(0.9954\), and \(0.9863\) in panel (**b**). The unconditioned state is shown as gray dots and its theoretical expectation as gray surface defined by dashed line. The fidelity value is \(f=0.9996\) in panel (**a**) and \(f=0.9999\) in panel (**b**). In general, the good agreement between the experimental data and the theoretical expectations validate the model used to describe the role played by the non-idealities of the employed detectors, namely the cross-talk effect, the non-unitary quantum efficiency and the possible imbalance between the two quantum efficiencies. We emphasize that the detection of subPoissonian states was achieved because, even in the presence of a limited quantum efficiency, the OCT probability is small enough to ensure that \(\eta>\eta_{th}\).