# Tailoring Asymmetric Lossy Channels to Test the Robustness of Mesoscopic Quantum States of Light
## 2 Materials and Methods
### Statistical Distributions of the Transmittance Coefficient
As an example, in Figure 1 we show a 3D plot of \(R\) as a function of \(\langle t\rangle\) and \(\sigma(t)\) for \(\eta=0.145\), \(\langle m\rangle=2.23\) and \(m_{\mu}=0.04\), which represent typical experimental values. As expected, the observation of nonclassical correlations becomes very difficult both at low values of \(\langle t\rangle\) and at values of \(\sigma(t)\) exceeding a threshold that depends on \(\langle t\rangle\). In the following, we consider two loss distributions that are involved in the propagation process of light through media. In particular, we deal with Gaussian and log-normal distributions. The Gaussian distribution of \(t\) can be expressed as  
\[P_{\rm g}(t)=\frac{1}{\sqrt{2\pi}\sigma_{0}}\exp\left[\frac{-(t-t_{0})^{2}}{2 \sigma_{0}^{2}}\right], \tag{5}\]  
where \(t_{0}\) and \(\sigma_{0}\) are the mean value and the standard deviation for t \(\in\) (\(-\infty\),\(+\infty\)), while the log-normal distribution is usually defined as  
\[P_{\rm In}(t)=\frac{1}{\sqrt{2\pi}\sigma_{\zeta}t}\exp\left[\frac{-[-\xi+\ln(t )]^{2}}{2\sigma_{\zeta}^{2}}\right], \tag{6}\]  
Figure 1: Theoretical expectation, according to Equation (4), of \(R\) (red surface) as a function of \(\langle t\rangle\) and of \(\sigma(t)\) for \(\eta=0.145\), \(\langle m\rangle=2.23\) and \(m_{\mu}=0.04\). The gray surface at \(R=1\) represents the boundary plane between classical and nonclassical correlations. in which \(\xi\in\Re\) is the so-called location parameter and \(\sigma_{\xi}>0\) is the scale parameter. Both parameters are linked to the mean value and the standard deviation of the distribution for \(t\in(0,\ +\infty)\): \(t_{0}=\exp[\xi+\sigma_{\xi}^{2}/2]\) and \(\sigma_{0}^{2}=(\exp[2\xi+\sigma_{\xi}^{2}])(-1+\exp[\sigma_{\xi}^{2}])\). For the specific application we are considering, the transmittance coefficient \(t\) is limited to the interval (0,1), which means that the distributions in Equations (5) and (6) must be properly normalized in the interval (0,1). The resulting probabilities can be expressed through closed formulas and the same holds for the first two moments of the distributions, \(\langle t\rangle\) and \(\sigma(t)\). In order to appreciate the differences between the two distributions, in Figure 2 we show some examples. We observe that, in general, at increasing the standard deviation, the discrepancies between Gaussian and log-normal distributions become more evident, since the log-normal distribution gets more asymmetric, while the Gaussian one remains symmetric. Moreover, comparing the two distributions for a given choice of standard deviation and mean value shows that for small mean values of \(t_{0}\) (see panels (a) and (b)) and large values of \(\sigma_{0}\) (see magenta curves) the log-normal distribution is confined in the region corresponding to small values of \(t\) (dashed line), while the Gaussian one is uniformly distributed all over the range (0,1) (solid line). On the contrary, for large mean values of \(t_{0}\) (see panels (c) and (d)) and large values of \(\sigma_{0}\) (see magenta curves) the Gaussian distribution has a longer tail towards small values of \(t\) than the log-normal one. At variance with these conditions, in all panels for small values of \(\sigma_{0}\) (see black curves) the two distributions are more confined and more similar. Before presenting the results of our investigation, we summarize the main features of the experimental implementation and explain the method used to prepare the data according to the specific distributions. Figure 2: Gaussian (solid lines) and log-normal (dashed lines) probability distributions of \(t\) in the interval (0, 1). The four panels (**a**–**d**) correspond to different choices of the mean value \(t_{0}\) of the distributions, as indicated in the label on top of each panel. Inside each panel, the different colors correspond to different choices of the standard deviation: \(\sigma_{0}=0.15\) is represented in black, while \(\sigma_{0}=0.7\) in magenta.