Generalized Bell-like inequality and maximum violation for multiparticle entangled Schrödinger cat states of spin $s$

This paper proposes a generalized Bell-like inequality (GBI) for multiparticle entangled Schrödinger cat states of arbitrary spin $s$. Based on quantum probability statistics, the GBI and violation are formulated in a unified manner with the help of a state density operator, which can be separated into local and nonlocal parts. The local part gives rise to the inequality, while the nonlocal part is responsible for the violation. The GBI is not violated at all by the quantum average, except the spin-$1/2$ entangled states. If the measuring outcomes are restricted in the subspace of the spin coherent state (SCS), namely, only the maximum spin values $\pm s$, the GBI is still meaningful for the incomplete measurement. With the help of SCS quantum probability statistics, it is proved that the violation of GBI can occur only for half-integer spins and not integer spins. Moreover, the maximum violation bound depends on the number parity of the entangled particles, which is $1/2$ for the odd particle numbers and 1 for the even numbers.