# Tailoring Asymmetric Lossy Channels to Test the Robustness of Mesoscopic Quantum States of Light
## 2 Materials and Methods  
### Statistical Distributions of the Transmittance Coefficient  
The transmission of light through a linear medium can be described by a wavelength- and polarization-dependent transmission coefficient. However, noise effects, such as absorption, depolarization, dephasing and scattering, can occur simultaneously, thus determining a variable transmittance coefficient or, more properly, a statistically-distributed transmittance coefficient. The situation is even more critical when the transmitted light is not represented by a single state but rather by a multipartite one. Indeed, in such a case, the different components of the state can experience different transmission efficiencies. This fact can determine the degradation of the correlations existing among the parties. It is thus crucial to quantify the amount of degradation in order to verify if the employed multipartite state is still useful for applications or not [32]. For instance, in the case of correlated bipartite systems, both classical and quantum, it is possible to get a fair estimation of the occurrence of degradation in terms of the noise reduction factor \(R\), which is defined as  
\[R=\frac{\sigma^{2}(n_{1}-n_{2})}{(\langle n_{1}\rangle+\langle n_{2}\rangle)}, \tag{1}\]  
in which \(\sigma^{2}(n_{1}-n_{2})\) is the variance of the distribution of the photon-number difference between the two parties, while \((\langle n_{1}\rangle+\langle n_{2}\rangle)\) is the shot-noise-level, that is the value of \(\sigma^{2}(n_{1}-n_{2})\) in the case of coherent states having mean values \(\langle n_{1}\rangle\) and \(\langle n_{2}\rangle\). Typically, the noise reduction factor is used to test the nonclassicality of quantum states of light, \(R<1\) being a sufficient condition for entanglement be expressed in terms of measurable quantities, such as the \"detected\" number of photons. As an example, in the case of multimode twin-beam states, the measured noise reduction factor reads [33]  
\[R=1-\frac{2\sqrt{\eta_{1}\eta_{2}}\sqrt{\langle m_{1}\rangle\langle m_{2} \rangle}}{\langle m_{1}\rangle+\langle m_{2}\rangle}+\frac{(\langle m_{1} \rangle-\langle m_{2}\rangle)^{2}}{\mu(\langle m_{1}\rangle+\langle m_{2} \rangle)} \tag{2}\]and the condition \(R<1\) proves the existence of sub-shot-noise correlations between the two parties. In Equation (2), \(\langle m_{1}\rangle\) and \(\langle m_{2}\rangle\) are the mean number of detected photons in the two arms of twin beam, \(\mu\) is the number of modes, and \(\eta_{1}\) and \(\eta_{2}\) are the detection efficiencies. In Ref. [17] we considered the presence of an asymmetric loss in the two arms of twin beam, and defined \(\langle m_{1}\rangle=\langle m\rangle=\eta\langle n\rangle\) and \(\langle m_{2}\rangle=\langle m\rangle t=\eta\langle n\rangle t\) (\(t\in\) (0,1)). The expression for \(R\) modifies as:  
\[R=1-\frac{2\eta t}{1+t}+\frac{(1-t)^{2}}{(1+t)}m_{\mu}, \tag{3}\]  
in which \(m_{\mu}=\langle m\rangle/\mu\) is the mean number of photons per mode. As stated above, in realistic situations, the asymmetric transmittance coefficient \(t\) is not constant but rather statistically distributed. In this case, by following the procedure presented in Ref. [17], it is still possible to find an analytic expression for the noise reduction factor \(R\), that is  
\[R=1-\frac{2\eta\langle t\rangle}{1+\langle t\rangle}+\frac{1-\langle t \rangle)^{2}}{1+\langle t\rangle}m_{\mu}+\left(\langle m\rangle+m_{\mu} \right)\frac{\sigma^{2}(t)}{1+\langle t\rangle}. \tag{4}\]  
We notice that in Equation (4) only the first two moments of the statistics of \(t\) are involved, namely \(\langle t\rangle\) and \(\sigma^{2}(t)\). This makes the investigation of the degradation of entanglement for twin-beam states particularly straightforward once the two moments of the distribution of \(t\) are known (or can be calculated).