with leads 1 and 2, as though terminals 3 and 4 were kept at a potential $eV$ with respect to 1 and 2. Our aim is to test the violation of the CH inequality given in Eq.(\[CHin\]) for such maximally entangled states.

When the angles $\theta_1$ and $\theta_2$ are parallel to each other, the scattering matrix of the two conductors, in the absence of spin mixing processes, can be written as: $$S= \left( \begin{array}{cc}
\hat{S}_{13} & 0 \\
0 & \hat{S}_{24}
\end{array} \right)\;
\label{S-ent}$$ where $$\hat{S}_{13}=\left (\begin{array}{cc}
\check{r}_{3} & \check{t}_{31} \\
\check{t}_{13} & \check{r}_{1}
\end{array} \right )= \left (\begin{array}{cccc}
r_{3 \uparrow} & 0 & t_{31 \uparrow} & 0 \\
0 &  r_{3 \downarrow} & 0 &  t_{31 \downarrow} \\
t_{13 \uparrow} & 0 & r_{1 \uparrow} & 0 \\
0 &  t_{13 \downarrow} & 0 &  r_{1 \downarrow} 
\end{array} \right ) .$$ Here $r_{j\sigma}$ ($t_{ij \sigma}$) is the probability amplitude for an incoming particle with spin $\sigma$ from lead $j$ to be reflected (transmitted in lead $i$). For a normal-metallic wire we set $t_{ij \uparrow}=t_{ij \downarrow}=\sqrt{T}$, $t_{ji \uparrow}=t_{ji \downarrow}=-\sqrt{T}$ and $r_{j\uparrow}=r_{j\downarrow}=\sqrt{1-T}$, where $T$ is the transmission probability. The expression for $\hat{S}_{24}$ is written analogously. For simplicity we will assume that 