the states $\{|z_3 \rangle,|-z_3 \rangle\}$ we get $$\rho_{12} = \langle z_3 |\Psi\rangle\langle\Psi|z_3 \rangle
+ \langle -z_3 |\Psi\rangle\langle\Psi|-z_3 \rangle ,$$ which can be identified with the density matrix (8) upon substitution of $|\Psi\rangle$ by the state vector (2).

The important point about the decomposition in Eq. (8) is that, as both $|\phi^{+}_{12}\rangle$ and $|\phi^{-}_{12}\rangle$ are product states, none of them violates Bell’s inequalities, and then the same will be true for the mixed state $\rho_{12}$ [@Barnett-Phoenix]. Indeed, one can easily find that the quantum prediction for the unconditional correlation function of the results of spin measurements on particles 1 and 2 along directions $\vec{e}_1$ and $\vec{e}_2$, respectively, is $$\begin{aligned}
E_{12}(\vec{e}_1,\vec{e}_2) &=& \text{Tr}[\rho_{12} \sigma(\vec{e}_1)\otimes
\sigma(\vec{e}_2)]  \nonumber  \\
&=& \pm \cos \vartheta_1 \cos \vartheta_2 ,\end{aligned}$$ where the matrix $\rho_{12}$ used in Eq. (9) stands for either one of expressions (7) or (8), and where the $+$ ($-$) sign applies for $\text{sgn}z_1 = \text{sgn}z_2$ ($\text{sgn}z_1 \neq \text{sgn}z_2$). So, in order to search for genuinely quantal correlations involving the pair of particles 1 and 2 it is necessary to consider the measurement results pertaining to either subensemble $+$ or $-$ separately. As a result, in the following we shall deal with the pure state $|\Psi^{+}_{12}\rangle$ 