P(Q_\alpha,\tau) \;
\label{noen}$$ (known as [*no-enhancement assumption*]{} ) is verified, it is possible to identify the variables appearing in Eq.(\[Clauser\]) as follows: $$\begin{aligned}
x&=& \frac{P^{\theta_1}(Q_1, \tau)}{P(Q_1,\tau)} ~~~ y=\frac{P^{\theta_2}
(Q_2,\tau)}{P(Q_2, \tau)} \; ,\nonumber\\
x'&=& \frac{P^{\theta'_1}(Q_1,\tau)}{P(Q_1,\tau)} ~~~ y'=\frac{P^{\theta'_2}
(Q_2, \tau)}{P(Q_2,\tau)} \; ,\label{variabili}\end{aligned}$$ $P^{\theta}(Q_\alpha,\tau)$ being the single terminal probability distribution in the presence of a analyzer. Eq.(\[Clauser\]) can then be rewritten in terms of probabilities by multiplying each side of the equation by $P(Q_1,\tau)P(Q_2,\tau) {\cal M}(\tau)d\tau$ and integrating over the space ${\cal T}$. Finally the following inequality is obtained $$\begin{aligned}
{\cal S}_{CH}&=&P^{\theta_1,\theta_2}(Q_1,Q_2)-P^{\theta_1,\theta'_2}(Q_1,Q_2)+
P^{\theta'_1,\theta_2}(Q_1,Q_2)+P^{\theta'_1,\theta'_2}(Q_1,Q_2)\;\nonumber\\
&-&P^{\theta'_1,-}(Q_1,Q_2)-P^{-,\theta_2}(Q_1,Q_2) \le 0 \; . 
\label{CHin}\end{aligned}$$ Eq.(\[CHin\]) is the CH inequality for the full counting statistics [^1], holding for all values of $Q_1$ and $Q_2$ which satisfy the no-enhancement assumption. We stress that the no-enhancement assumption, upon which Eq.(\[CHin\]) is based, it is not satisfied in general like its optical version. The quantities that we have to compare are probability distributions, so that Eq.(\[noen\]) must be checked over the whole range of $Q$. For a fixed time $t$ and a given mesoscopic system, hence for a given scattering matrix and incident particle state, the no-enhancement assumption is valid only in some range of values of $Q$. In particular, different sets 