only for appropriate values of $M$, $T$ and $Q_1$ or $Q_2$. For example, for a given observation time $t$ ([*i.e.*]{} a given $M$) and a given value of $Q$, CH inequality can be tested only for transmission $T$ less than a maximum value given by the expression $$T_{\text{max}}=\frac{2^{\frac{Q_i}{M-Q_i}}-1}{2^{\frac{Q_i}{M-Q_i}}-\frac{1}{2}} .
\label{condition}$$ At the edge of the distribution ($Q_i=M$) the no-enhancement assumption is satisfied for every $T$. The window of allowed $Q_i$ values where the no-enhancement assumption is satisfied gets wider on approaching the tunneling limit. For large $M$, $T_{\text{max}}\simeq  2 (\log2) \frac{Q_i}{M}$. The previous inequality can be also interpreted as a limit for the allowed measuring time given a setup at disposal. Alternatively, given a certain transmission, the no-enhancement assumption is verified for points of the distribution such that: $$\frac{Q_i}{M}\geq \frac{\log\frac{1-T/2}{1-T}}{\log 2+\log\frac{1-T/2}{1-T}} .
\label{ultimo}$$

The various probabilities needed to define ${\cal S}_{CH}$ are collected in Appendix \[PR\]. However, it is useful to note here that the joint probabilities with a single analyzer are factorized: $$\begin{aligned}
\nonumber
P^{\theta_1,-}(Q_1,Q_2)=P^{\theta_1}(Q_1) P(Q_2)\\
P^{-,\theta_2}(Q_1,Q_2)=P(Q_1) P^{\theta_2}(Q_2) ,\end{aligned}$$ while joint probabilities with two analyzers are not factorized. Furthermore, all such probabilities have a common factor, $T^{Q_1+Q_2}/2^M$, which leads to an exponential suppression for large $M$ and $Q_1+Q_2$. We 