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

Abstract

In terms of quantum probability statistics, the Bell inequality (BI) and its violation are extended to spin-[Formula: see text] entangled Schrödinger cat state (called the Bell cat state) with both parallel and antiparallel spin-polarizations. Except the spin-1/2, the BI is never ever violated by the Bell cat states with the measuring outcomes including entire Hilbert space. If, on the other hand, measuring outcomes are restricted in the subspace of spin coherent state (SCS), a universal Bell-type inequality (UBI), [Formula: see text], is formulated in terms of the local realistic model. We observe a spin parity effect 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 nontrivial Berry phase between the SCSs 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 [Formula: see text], which is valid for arbitrary half-integer spin-[Formula: see text] states.