# Measuring outcome correlation for spin-s Bell cat-state and geometric phase induced spin parity effect

## TL;DR

This paper extends Bell inequality analysis to spin-$s$ entangled states, revealing a spin parity effect where violations occur only for half-integer spins due to geometric phase differences.

## Contribution

It introduces a universal Bell-type inequality for spin-$s$ states and uncovers a spin parity effect linked to Berry phase and geometric phase differences.

## Key findings

- UBI is violated only by half-integer spin states.
- Maximum violation bound of UBI is 1 for all half-integer spins.
- Violation is caused by non-trivial Berry phase in half-integer spins.

## Abstract

In terms of quantum probability statistics the Bell inequality (BI) and its violation are extended to spin-$s$ entangled Schr\"{o}dinger cat-state (called the Bell cat-state) with both parallel and antiparallel spin-polarizations. The BI is never ever violated for the measuring outcome probabilities evaluated over entire two-spin Hilbert space except the spin-$1/2$ entangled states. A universal Bell-type inequality (UBI) denoted by $p_{s}^{lc}\leq0$ is formulated with the local realistic model under the condition that the measuring outcomes are restricted in the subspace of spin coherent states. A spin parity effect is observed that the UBI can be violated only by the Bell cat-states of half-integer but not the integer spins. The violation of UBI is seen to be a direct result of non-trivial Berry phase between the spin coherent states of south- and north-pole gauges for half-integer spin, while the geometric phase is trivial for the integer spins. A maximum violation bound of UBI is found as $p_{s}^{\max}$=1, which is valid for arbitrary half-integer spin-$s$ states.