# Tailoring Asymmetric Lossy Channels to Test the Robustness of Mesoscopic Quantum States of Light
## 3 Results  
At variance with Ref. [17], in which we focused our attention on the possibility of keeping observing nonclassicality in the presence of an asymmetric loss between the two parties of the twin-beam states, here we aim at finding the limits imposed by the parameters that describe the transmittance statistics, both for the Gaussian and the log-normal distributions. According to Equation (4), the noise reduction factor is a function of the first two moments of the distribution of \(t\), independent of the considered distribution. This means that by choosing the same mean value \(\langle t\rangle\) and the same standard deviation \(\sigma(t)\), the Gaussian and the log-normal distributions act in the same way. On the contrary, if we consider the values of \(t_{0}\) and \(\sigma_{0}\) over the entire domains, we expect different results. To better emphasize this point, in panel (a) of Figure 4 we show, as contour plot, the difference \(\langle t\rangle_{\text{GAUSS}}-\langle t\rangle_{\text{LOG}}\) between the mean value \(\langle t\rangle\) of the normalized Gaussian distribution and that of the  
Figure 3: (Color online) Sketch of the experimental setup. See the text for details. normalized log-normal distribution as a function of \(t_{0}\) and \(\sigma_{0}\), while in panel (b) of Figure 4 we do the same for the standard deviation \(\sigma(t)\), namely we plot the difference \(\sigma(t)_{\rm GAUSS}-\sigma(t)_{\rm LOG}\). In panel (a) (panel (b)), the pink region corresponds to values of \(\langle t\rangle_{\rm GAUSS}-\langle t\rangle_{\rm LOG}\left(\sigma(t)_{\rm GAUSS }-\sigma(t)_{\rm LOG}\right)\) larger than 0, the gray region to values lower than 0, and the black-dashed line dividing the pink region from the gray one corresponds to the condition \(\langle t\rangle_{\rm GAUSS}-\langle t\rangle_{\rm LOG}=0\) (panel (a)), and to the condition \(\sigma(t)_{\rm GAUSS}-\sigma(t)_{\rm LOG}=0\) (panel (b)). As it can be noticed by comparing the two panels, it is not possible to find a set of \(t_{0}\) and \(\sigma_{0}\) values corresponding to equal values of \(\langle t\rangle\) and \(\sigma(t)\) for the two distributions, unless one considers sparse choices of \(t_{0}\) and \(\sigma_{0}\) for \(\sigma_{0}<0.05\). For the above reasons, a direct comparison between the distributions could be not essential. On the contrary, it can be more interesting, for each one of the two distributions separately, to investigate the survival of nonclassical correlations as a function of its first two moments. In particular, it is straightforward to deal with the expressions defined over the entire domain and their corresponding moments by considering the link among \(\langle t\rangle\) and \(\sigma(t)\) in Equation (4) to \(t_{0}\) and \(\sigma_{0}\). First of all, we consider the case of the Gaussian distribution. In Figure 5 we show the noise reduction factor as a function of the standard deviation \(\sigma_{0}\) for different values of \(t_{0}\). Even if, in general, \(\sigma_{0}\) can assume values larger than 1, in our analysis we explored only values in the interval (0,1) since they are sufficient to observe the degradation of nonclassicality. The data, shown as colored dots + error bars, are superimposed to the theoretical fitting curves that can be obtained according to Equation (4), in which we fix the values of \(\langle m\rangle\) and \(\eta\) and leave \(m_{\mu}\) and \(t_{0,\rm FIT}\) as fitting parameters. In particular, we set \(\langle m\rangle=2.23\), which is the maximum value of the mean number of photons detected in the variable arm, and \(\eta=0.145\), which is the quantum efficiency of the detection chain obtained as \(\eta=1-R_{MIN}\), being \(R_{MIN}\) the value of the noise reduction factor corresponding to \(t=1\). In the fitting procedure, we leave \(t_{0,\rm FIT}\) as a free fitting parameter to take into account the possibility that the preparation of the distribution of \(t\) is not ideal due to the discrete values of \(t\) at our disposal.