Violation of Bell-type Inequalities on Mutually-commuting von Neumann Algebra Models of Entanglement Swapping Networks

TL;DR

This paper models entanglement swapping networks using mutually-commuting von Neumann algebras, establishing Bell-type inequalities and their violations to explore algebraic structures and classify von Neumann algebra types.

Contribution

It generalizes bipartite Bell inequality models to multipartite networks within the von Neumann algebra framework, revealing structural conditions for violations and classification.

Findings

Derived bounds for Bell-type inequalities based on von Neumann algebra structures.

Identified algebraic conditions necessary for Bell inequality violations.

Maximal Bell violation helps classify the type of underlying von Neumann algebras.

Abstract

Violation of Bell inequalities in bipartite systems represented by mutually-commuting von Neumann algebras has pioneered the study of vacuum entanglement in algebraic quantum field theory. It is unexpected that the maximal violation of Bell inequality can discover algebraic structures. In the paper, we establish the mutually-commuting von Neumann algebra model for entanglement swapping networks and Bell-type inequalities on this model. It generalizes the bipartite case to the ternary case. These algebras are all general von Neumann algebras, which provide a natural perspective to investigate Bell nonlocality in quantum networks in the infinitely-many-degree-of-freedom setting. We determine various bounds for Bell-type inequalities based on the structure of von Neumann algebras, and identify the algebraic structural conditions required for their violation. Finally, we show that the maximal violation of Bell-type inequalities in entanglement swapping networks can be used to determine partially the type classification of the underlying von Neumann algebras.