Multisetting Bell inequalities for $N$ spins-1 avoiding KS contradiction

TL;DR

This paper derives new multisetting Bell inequalities for N spins-1 systems that avoid Kochen-Specker contradictions, demonstrating robust violations that increase with the number of subsystems using only spin observables.

Contribution

It introduces a method to construct Bell inequalities for spins-1 systems that bypass known algebraic contradictions, expanding the scope of Bell tests.

Findings

Derived multisetting Bell inequalities for spins-1 systems.

Showed inequalities are robustly violated by quantum mechanics.

Violations increase with the number of subsystems.

Abstract

Bell's theorem for systems more complicated than two qubits faces a hidden, as yet undiscussed, problem. One of the methods to derive Bell's inequalities is to assume existence of joint probability distribution for measurement results for all settings in the given experiment. However for spins-1, one faces the problem that eigenvalues of observables do not allow a consistent algebra if one allows all possible settings on each side (Bell 1966 contradiction), or some specific sets (leading to a Kochen-Specker 1967 contradiction). We show here that by choosing special set of settings which never lead to inconsistent algebra of eigenvalues, one can still derive multisetting Bell inequalities, and that they are robustly violated. Violation factors increase with the number of subsystems. The inequalities involve only spin observables, we do not allow all possible qutrit observables, still the violations are strong.