SU(1,1) parity and strong violations of a Bell inequality by entangled Barut–Girardello coherent states

We study the violations of the Bell–Clauser–Horne–Shimony–Holt inequality for entangled SU(1,1) Barut–Girardello coherent states. As in a previous paper where violations of the inequality were studied for entangled SU(1,1) coherent states of the Perelomov form [Phys. Rev. A.93, 042104 (2016)PLRAAN1050-294710.1103/PhysRevA.93.042104], we choose as our observable the SU(1,1) parity operator, though here we discuss the physical meaning of the operator for single-mode and two-mode bosonic realizations of the su(1,1) Lie algebra. We show that the SU(1,1) parity operator is not the same as the usual photon number parity operator but rather is a form of higher-order photon number parity. Distant observers Alice and Bob perform on each of their subsystems non-compact SU(1,1) transformations characterized by hyperbolic angles, followed by measurements of the SU(1,1) parity operators. We find strong violations of the inequality over a wide range of parameters, and these violations are stronger than those that can be obtained by the entangled Perelomov SU(1,1) coherent state.