# Tailoring Asymmetric Lossy Channels to Test the Robustness of Mesoscopic Quantum States of Light
## 1 Introduction  
Since the seminal paper in which [PERSON] and [PERSON] dealt with the transmission of quantum states and cryptographic keys through 0.3-m-long free-space air [1], Quantum Communication over long distances has received a lot of interest. In the past two decades, many experiments have been performed using both optical fibers [2; 3; 4] and free-space [5; 6; 7] channels. Starting from the successful implementation of ground-to-ground atmospheric links [8; 9], some most recent experiments have also involved a satellite link [10; 11; 12]. Despite all these results, the development of a real global communication network in free-space propagation is still prevented by the atmospheric turbulence, which acts as a temporal and spatial variation of the air refraction index, thus varying the transmittance of the links in a turbulent way. In order to understand the behavior of quantum states of light propagating through atmospheric links, a deep investigation of quantum channels is required [13]. In this respect, quite recent works have introduced different fluctuation loss models capable of accurately describe some experimental results [14; 15]. At the same time, it is also important to investigate which kinds of quantum states and nonclassical features are more robust against atmospheric fluctuations and can survive under specific conditions [16]. Until now, most experiments have been implemented at the single-photon level. Recently, we have realized an experiment involving mesoscopic twin-beam states, in which signal and idler were affected by different amounts of loss distributed according to specific statistical distributions [17]. In particular, we have investigated how nonclassicality changes as a function of the mean value of the distributions for fixed values of their standard deviation. Here we face the problem from a different point of view, that is weconsider the case in which the standard deviation of the distribution is varied and the mean value of the distribution is fixed. At variance with the previous paper, we deeply analyze the behavior of nonclassicality by choosing worse and worse situations, down to very low and very noisy transmission-efficiency values. For all the chosen values of the parameters, we consider two possible transmittance distributions, namely the log-normal distribution [18,19], which is typically used to describe very turbulent transmission channels, and the Gaussian distribution, which can be exploited to model free-space channels under specific weather conditions [20]. To investigate the nonclassicality of the generated twin beams, which are entangled in the number of photons, we consider the noise reduction factor (\(R\)). Indeed, it has been demonstrated that \(R<1\) represents a sufficient criterion for entanglement in the case of bipartite Gaussian states [21]. Actually, many other nonclassicality criteria exist and have been proposed over the years [22, 23, 24, 25, 26, 27]. However, not all of them can be easily applied to any experimental situation. Some of them, such as those based on separability criteria, require the full reconstruction of the state under investigation. On the contrary, the noise reduction factor can be calculated from experimental data in a more direct way, also because it can be defined in terms of measurable quantities [28] (see the next section). The obtained results shed light on the different behavior of nonclassicality according to the chosen distribution of the transmission efficiency and on the evolution of such a behavior as a function of the different parameters. To be more exhaustive, in this work we also investigate the role played by the mean number of photons per mode of the employed quantum states in the entanglement degradation process.  
Our analysis can give some useful hints not only for the exploitation of quantum states in communication protocols, but also for their application in different contexts, such as for imaging protocols [29, 30, 31].