# Tailoring Asymmetric Lossy Channels to Test the Robustness of Mesoscopic Quantum States of Light
## 4 Discussion  
The different behavior of the mean number of photons per mode for the two distributions can be ascribed to the fact that, when the mean number of photons is varied in one arm of the twin beam, also the number of modes changes. Indeed, the number of modes we obtain from the first two moments of the light statistics is an \"effective\" one describing the multimode state as the tensor product of \(\mu\) equally-populated single-mode states. This is just a useful approximation since in the real experiment the different modes are differently populated [35]. For this reason, the attenuation of light by a filter (the HWP followed by the PBS is equivalent to such a condition) can vary the number of effective modes since some of the real modes can be so attenuated to go below the detection threshold. In particular, we experimentally observed that the number of effective modes monotonically increases at increasing the values of the mean number of photons. The variability of the number of modes is amplified when the data are combined according to specific distributions of \(t\). As discussed above, this is more visible in the case of log-normal distribution due to its nontrivial shape at different mean values. To better investigate the role played by the mean number of photons per mode in the calculation of the noise reduction factor for the two considered distributions, we theoretically study the behavior of \(R\) as a function of \(\sigma_{0}\) by properly fixing the parameters appearing in Equation (4). In particular, we set \(\left\langle m\right\rangle=2.23\), \(\eta=0.145\) and consider two possible values of \(m_{\mu}\), \(0.1\) and \(0.04\). The resulting expressions for \(R\) are plotted in Figure 6.  
Figure 6: Theoretical values of the noise reduction factor as a function of the standard deviation \(\sigma_{0}\) of the Gaussian (panels **(a,b)**) and log-normal (panels **(c,d)**) distributions for different values of \(t_{0}\). For all the shown curves we set \(\left\langle m\right\rangle=2.23\), \(\eta=0.145\). In panels **(a,c)**\(m_{\mu}=0.1\), while in panel **(b,d)**\(m_{\mu}=0.04\). In each panel, the pink curve corresponds to \(t_{0}=0.4\), the magenta curve to \(t_{0}=0.5\), the green curve to \(t_{0}=0.6\), the blue curve to \(t_{0}=0.7\), the black curve to \(t_{0}=0.8\) and the red curve to \(t_{0}=0.9\). We note that the choice \(\langle m\rangle=2.23\), and \(m_{\mu}=0.1\) corresponds to a twin-beam state with \(\sim\)22 modes, while \(\langle m\rangle=2.23\), and \(m_{\mu}=0.04\) corresponds to a twin-beam state with \(\sim\)56 modes, which represents a more reliable experimental condition than the first choice. The curves shown in panels (a) and (b) are for Gaussian distributions, while those in panels (c) and (d) are for log-normal distributions. In general, the plots resemble the experimental behavior of the plots in Figure 5. However, we notice that depending on the choice of \(m_{\mu}\), the value of \(\sigma_{0}\) at which the noise reduction factor is equal to 1, namely the boundary between classical and nonclassical correlations, changes. In particular, the lower the value of \(m_{\mu}\), the larger the threshold value of \(\sigma_{0}\). Indeed, according to Equation (4), the lower the value of \(m_{\mu}\), the lower the value of \(R\). To better investigate this result, in Figure 7 we plot the values of \(\sigma_{0}\) at which, for fixed choices of \(\langle m\rangle\) and \(\eta\) and for 4 possible choices of \(m_{\mu}\), the theoretical value of \(R\) is equal to 1 as a function of \(t_{0}\). We show the results in the case of Gaussian distributions in panel (a) and those for log-normal distributions in panel (b). In general, we can see that the threshold values of \(\sigma_{0}\) increase at increasing values of \(t_{0}\) with different slopes for the two distributions. Moreover, for log-normal distributions of \(t\) the growth is more rapid than for Gaussian distributions.