&& - \cos\vartheta^{\prime}_{1} \cos\vartheta^{\prime}_{2}]
\pm z_3 (c_1 c_2 /p_{\pm}) [\sin\vartheta_1 \sin\vartheta_2 \sin\vartheta_3 \nonumber   \\
&& \times \cos (\varphi_1 +\gamma \varphi_2+z_1 z_3 \varphi_3) +\sin\vartheta_1 \sin\vartheta^{\prime}_{2} \sin\vartheta_3 \nonumber  \\
&& \times \cos (\varphi_1 +\gamma\varphi^{\prime}_{2} +z_1 z_3 \varphi_3) 
+ \sin\vartheta^{\prime}_{1} \sin\vartheta_2 \sin\vartheta_3 \nonumber  \\
&& \times \cos (\varphi^{\prime}_{1} +\gamma\varphi_2 +z_1 z_3 \varphi_3)  
- \sin\vartheta^{\prime}_{1} \sin\vartheta^{\prime}_{2} \sin\vartheta_3 \nonumber  \\
&& \times \cos (\varphi^{\prime}_{1} +\gamma\varphi^{\prime}_{2} +z_1 z_3 \varphi_3)] | > 2.\end{aligned}$$ It should be realized that, by suitable choosing the $z$ axis for each of the particles, any entangled state for two spin-$\frac{1}{2}$ particles can be put into the form displayed by either one of Eqs. (5a) or (5b). So the above condition (13) for the violation of the CHSH inequality turns out to be completely general as far as pure states are concerned. For the special case in which $\vartheta^{\prime}_{1} = \vartheta_1$, $\vartheta^{\prime}_{2} = \vartheta_2$, $\varphi^{\prime}_{1} = \varphi_1 +\pi/2$, $\gamma\varphi^{\prime}_{2} = \gamma\varphi_2 +\pi/2$, and $\varphi_1 +\gamma\varphi_2 +z_1 z_3 \varphi_3 = 3\pi/4 +n\pi$, condition (13) simplifies to $$|\gamma \cos\vartheta_1 \cos\vartheta_2 \pm \mu z_3 (c_1 c_2 /p_{\pm}) \sqrt{2}
\sin\vartheta_1 \sin\vartheta_2 \sin\vartheta_3 | >1,$$ where $\mu$ is a sign factor equal to $+1$ ($-1$) for $n$ odd (even). From expression (14) it follows that, as long as 