Maximal violations of a Bell inequality by entangled spin-coherent states

We study the violation of the Bell-Clauser-Horne-Shimony-Holt (Bell-CHSH) inequality for two-spin systems, of arbitrary spins $j_1$ and $j_2$, prepared in an entanglement of spin-coherent states of each of the spins. We show that the Bell-CHSH inequality is quite robustly violated for a wide range of values of the parameters that specify the spin-coherent states, and, for a particular choice of these parameters, maximal violations are obtained. That is, the violations can reach the Tsirelson bound, $2\sqrt{2}$, for any choices of the spins.

Figure 1

Diagram showing the arrangements of the auxiliary angles ${\Theta }_1$, ${\Theta }_2$, ${\Theta }_3$, and ${\Theta }_4$. We make the choices ${\Theta }_1-{\Theta }_3=\varphi$, ${\Theta }_1-{\Theta }_4=-\varphi$, ${\Theta }_2-{\Theta }_3=-\varphi$, such that ${\Theta }_2-{\Theta }_4=-3\varphi$.