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arXiv:2604.17765

Mutually-commuting von Neumann algebra models of quantum networks and violation of Bell-type inequalities

Shuyuan Yang, Jinchuan Hou, Kan He

quant-phmath-phmath.OASubmitted Apr 20, 2026• Announced Apr 21, 2026• 0 citations
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Abstract

Employing mutually-commuting von Neumann algebras to represent the algebra of observables on quantum systems provides a framework for studying quantum information theory in systems with infinite degrees of freedom and quantum field theory, yielding many profound results that differ from non-relativistic quantum systems. In this paper, we establish a mutually-commuting von Neumann algebra model of quantum networks with arbitrary structures. We derive Bell-type inequalities on this model, and determine various bounds for Bell-type inequalities based on the structure of underline von Neumann algebras, and identify the algebraic structural conditions required for their violation. The conditions on the algebraic structure of observables for maximal violation of Bell-type inequalities, which we discovered in the context of von Neumann algebra models, can in turn guide the search for measurements in the non-relativistic setting.

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Employing mutually-commuting von Neumann algebras to represent the algebra of observables on quantum systems provides a framework for studying quantum information theory in systems with infinite degrees of freedom and quantum field theory, yielding many profound results that differ from non-relativistic quantum systems. In this paper, we establish a mutually-commuting von Neumann algebra model of quantum networks with arbitrary structures. We derive Bell-type inequalities on this model, and determine various bounds for Bell-type inequalities based on the structure of underline von Neumann algebras, and identify the algebraic structural conditions required for their violation. The conditions on the algebraic structure of observables for maximal violation of Bell-type inequalities, which we discovered in the context of von Neumann algebra models, can in turn guide the search for measurements in the non-relativistic setting.
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.
We continue our study of Bell's inequalities and quantum field theory. It is shown in considerably broader generality than in our previous work that algebras of local observables corresponding to complementary wedge regions maximally violate Bell's inequality in all normal states. Pairs of commuting von Neumann algebras that maximally violate Bell's inequalities in all normal states are characterized. Algebras of local observables corresponding to tangent double cones are shown to maximally violate Bell's inequalities in all normal states in dilatation-invariant theories, in free quantum field models, and in a class of interacting models. Further, it is proven that such algebras are not split in any theory with an ultra-violet scaling limit On montre que les algebres d'observables locales correspondant a des domaines en coins violent au maximum les inegalites de Bell dans tous les etats normaux. On caracterise les paires d'algebres de Von Neumann qui commutent et qui violent maximalement les inegalites de Bell. On montre que les algebres d'observables correspondant a des cones doubles tangents violent aussi maximalement les inegalites de Bell dans tous les etats normaux des theories invariantes par dilatation, dans les modeles de champs libres et dans des modeles en interaction. On montre que ces algebres ne sont «split» dans aucune theorie avec une limite d'echelle UV
This paper explores the intricate connections between various algebraic structures—Banach algebras, C*-algebras, and von Neumann algebras—and their profound implications for the theory of quantum entanglement and quantum information. We establish a comprehensive framework that bridges the abstract concepts of functional analysis with the concrete demands of quantum mechanics. Beginning with the foundational properties of Banach algebras, we progressively introduce C*-algebras as a natural setting for quantum observables and states, followed by von Neumann algebras, which are essential for describing quantum measurement and infinite-dimensional quantum systems. A central focus is to delineate how quantum entanglement, a cornerstone resource in quantum information science, can be rigorously characterized and quantified within this hierarchical algebraic framework. We integrate recent game-theoretic approaches to quantum information, providing new perspectives on quantum relative entropy, mutual information, and channel capacity. By demonstrating how the rich structure of C*-algebras acts as a pivotal bridge, we illuminate the mathematical coherence underlying the transition from general algebraic concepts to the specialized domain of quantum information processing, offering a unified view of quantum phenomena. The work also delves into the operational justifications for these algebraic descriptions, particularly in the context of infinite quantum systems.
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One of the essential features of quantum theory is the noncommutativity of observables. Moreover, under the non-contextual realistic interpretations of quantum theory, all quantum observables must commute {[J. D. Malley, Phys. Rev. A {\\bf 69}, 022118 (2004)]}. Experimental realizations of violations of Bell inequality say the noncommutativity of observables in an entangled state. Here we show that commutativity can be tested experimentally using uncorrelated states, with no dependence on the algebraic structure of the observables under study. The test is derived from the bounds of an inequality valid for all commuting observables. For nnn spin-1/2 systems we show that an nnn-partite uncorrelated state violates the inequality by an amount that grows exponentially with nnn.
Bell inequalities bound the strength of classical correlations between observers measuring on a shared physical system. However, studies of physical correlations can be considered beyond the standard Bell scenario by networks of observers sharing some configuration of many independent physical systems. Here, we show how to construct Bell-type inequalities for correlations arising in any tree-structured network, i.e., networks without cycles. This is achieved by an iteration procedure that in each step allows one to add a branch to the tree-structured network and construct a corresponding Bell-type inequality. We explore our inequalities in several examples, in all of which we demonstrate strong violations from quantum theory.