Generalized Bell-like inequality and maximum violation for multiparticle entangled Schr\"{o}dinger-cat-states of spin-s

TL;DR

This paper develops a generalized Bell-like inequality for multiparticle entangled states of arbitrary spin, revealing that violations occur only for half-integer spins and depend on the number parity of particles, with maximum violations quantified.

Contribution

It introduces a unified formulation of the GBI using quantum probability and density operators, extending Bell inequalities to arbitrary spin multiparticle entangled states and analyzing their violation conditions.

Findings

GBI not violated by quantum averages except for spin-1/2 states

Violations occur only for half-integer spins, not integer spins

Maximum violation depends on particle number parity, 1/2 for odd, 1 for even

Abstract

This paper proposes a generalized Bell-like inequality (GBI) for multiparticle entangled Schr\"{o}dinger-cat--states of arbitrary spin-$s$. Based on quantum probability statistics the GBI and violation are formulated in an unified manner with the help of state density operator, which can be separated to local and non-local parts. The local part gives rise to the inequality, while the non-local part is responsible for the violation. The GBI is not violated at all by quantum average except the spin-$1/2$ entangled states. If the measuring outcomes are restricted in the subspace of 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 but not integer spins. Moreover, the maximum violation bound depends on the number parity of entangled particles, that it is $1/2$ for the odd particle-numbers while $1$ for even numbers.