Violation of Quantum Bilocal Inequalities on Mutually-Commuting von Neumann Algebra Models

Differently from the non-relativistic quantum mechanics, the violation of Bell inequalities in quantum field theory depends more on the structure of observable algebras (typically type III von Neumann algebras) rather than the choice of specific quantum states. Therefore, studying the violation of Bell inequalities based on the von Neumann algebraic framework often reveals information about the algebraic structure. In this paper, we employ three mutually-commuting von Neumann algebras to characterize quantum entanglement swapping networks, and establish Bell-like inequalities thereon, commonly referred to as bilocal inequalities. We investigate the algebraic structural conditions under which bilocal inequalities are satisfied or violated on the generated algebra of these three von Neumann algebras. Furthermore, the conditions for maximal violation of the inequalities can be utilized to infer the structural information of von Neumann algebras in reverse. Our results not only utilize the violation of bilocal inequalities to reveal the structural properties of von Neumann algebras, but can also be applied to quantum mechanics and quantum field theory.