\theta_2}{2}$. This allows us, without loss of generality, to define an angle $\Theta$ such that $2\Theta=\theta_1\pm \theta_2=\theta'_1\pm \theta_2=
\theta'_1\pm \theta'_2=(\theta_1\pm \theta'_2)/3$. As a result Eq.(\[CHin\]) takes the form: $${\cal S}_{CH}=3P_{1,2}^{\Theta}(Q_1,Q_2)-P_{1,2}^{3\Theta}(Q_1,Q_2)-P_{1,-}(Q_1,Q_2)-P_{-,2}(Q_1,Q_2)
\le 0$$ where $P_{1,2}^{\Theta}=P^{\theta_1,\theta_2}$ and $P_{1,-}=P^{\theta_1,-}$. It is useful to define the reduced quantity $\overline{{\cal S}}_{CH}={\cal S}_{CH}/(T^{2M}/2^M)$ which is plotted in Fig. \[Srid:Q=M\] as a function of $\Theta$ for different values of $M$ (note that since $P^{\theta_1,-}(M,M)=(T^{2M}/2^M)$, $\overline{{\cal S}}_{CH}$ is nothing but ${\cal S}_{CH}/P^{\theta_1,-}(M,M)$). The violation occurs for every value of $M$ in a range of angles around $\Theta=\pi/2$ (note that ${\cal S}_{CH}$ is symmetric with respect to $\pi/2$). The range of angles for which $\overline{{\cal S}}_{CH}$ is positive shrinks with increasing $M$, while the maximum value of $\overline{{\cal S}}_{CH}$ decreases very weakly with $M$ (more precisely, $\overline{{\cal S}}_{CH}^{\text{max}}\propto 1/M$). This means that the effect of the factor $T^{2M}/2^M$ on the value of ${\cal S}_{CH}$ is exponentially strong, making the violation of the CH inequality exponentially difficult to detect for large $M$ and $Q_1=Q_2=M$. The weakening of the violation is mainly due to the suppression of the joint probabilities. As we shall show later, by optimizing all the parameters it is yet possible to eliminate this exponential suppression.

Let us 