or $|\Psi^{-}_{12}\rangle$, rather than with the mixed state $\rho_{12}$.

It is worth noting that the inability of the entangled three-particle state (2) to yield nonlocal correlations involving the unconditional measurement results for two of the particles can be traced back to the orthogonality of the states $|z_i\rangle$ and $|-z_i\rangle$, for each $i=1,2,$ and 3. So we can say that the lack of such nonlocal, unconditional two-particle correlations indeed constitutes a significant feature inherent in the class of triorthogonal states. In Sec. IV we shall generalize this result to the class of $n$-orthogonal states.

Maximal violation of the CHSH inequality: necessary conditions
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The Clauser-Horne-Shimony-Holt (CHSH) form of the Bell inequality is [@CHSH] $$|\langle B_{\text{CHSH}}\rangle | \leq 2 \, ,$$ where $\langle B_{\text{CHSH}}\rangle$ denotes the expectation value of the Bell operator [@BMR] $$\begin{aligned}
B_{\text{CHSH}} &\equiv & \sigma(\vec{e}_1) \otimes [\sigma(\vec{e}_2) + \sigma(\vec{e}\,%
^{\prime}_{2})] + \sigma(\vec{e}\,^{\prime}_{1}) \nonumber  \\
&& \otimes \, [\sigma(\vec{e}_2) - \sigma(\vec{e}\,^{\prime}_{2})] \, ,\end{aligned}$$ with $\vec{e}_i$ and $\vec{e}\,^{\prime}_{i}$ denoting two different alternative directions for spin measurements on particle $i$. As long as $\vartheta_3 \neq n\pi$, either two-particle state (5a) or (5b) will be able to violate the CHSH inequality for a suitable choice of measurement directions $\vec{e}_1$, $\vec{e}\,^{\prime}_{1}$, $\vec{e}_2$, and $\vec{e}\,^{\prime}_{2}$ 