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arXiv:2603.01466

Violation of Bell-type Inequalities in Entanglement Swapping Networks Represented by Mutually-commuting von Neumann Algebras

Bingke Zheng, Shuyuan Yang, Jinchuan Hou, Kan He

math.FAmath-phmath.OAquant-phSubmitted Mar 2, 2026• Announced Mar 3, 2026• 0 citations• Revised Apr 9, 2026 (v3)
v1 · Mar 2, 2026v2 · Apr 1, 2026v3 · Apr 9, 2026v4 · Apr 10, 2026v5 · Apr 15, 2026
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Abstract

Violation of Bell inequalities in bipartite systems represented by mutually-commuting von Neumann algebras has pioneered the study of vacuum entanglement, and linked Bell nonlocality to the locality conditions in algebraic quantum field theory. In the paper, we establish the mutually-commuting von Neumann algebra model for entanglement swapping networks and Bell-type inequalities on this model. 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. The most unexpected result is that all normal network states can lead to the violation of these inequalities. This demonstrates that the violation of Bell-type inequalities is determined intrinsically by the structural properties of these algebras. Finally, we show the application of the aforementioned conclusions into the quantum field theory.

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Violation of Bell inequalities in bipartite systems represented by mutually-commuting von Neumann algebras has pioneered the study of vacuum entanglement, and linked Bell nonlocality to the locality conditions in algebraic quantum field theory. In the paper, we establish the mutually-commuting von Neumann algebra model for entanglement swapping networks and Bell-type inequalities on this model. 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. The most unexpected result is that all normal network states can lead to the violation of these inequalities. This demonstrates that the violation of Bell-type inequalities is determined intrinsically by the structural properties of these algebras. Finally, we show the application of the aforementioned conclusions into the quantum field theory.
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.
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
In an entanglement swapping scenario, if two sources distributing entangled states between three parties are independent, local correlations lead to a different kind of inequality than the standard Bell inequalities, known as network local models. A highly demanding task is to find out a way to involve many players nontrivially in a quantum network since measurements, in general, disturb the system. To this end, we consider here an alternative way of sharing network nonlocality when two observers initially share a maximally entangled, a nonmaximally entangled and a noisy entangled state. We report that by employing unsharp measurements performed by one of the observers, at most six pairs can sequentially demonstrate the violation of bilocal inequality while a maximum of two pairs of observers can exhibit bi-nonlocality when both the observers perform unsharp measurements. We also find the critical noise involved in unsharp measurements in each round to illustrate the bi-nonlocality for a fixed shared entangled state as a resource. We also connect the entanglement content of the shared state, quantified via von-Neumann entropy for pure states and entanglement of formation for Werner states with the maximum number of rounds showing violation of bilocal inequality. By considering elegant joint measurement at the node such that the entanglement content in the elements of the joint measurement by the third party gets reduced, we observe that the maximum number reduces to two sequential sharing of bi-nonlocality even for the maximally entangled state when the settings at each side are taken to be three and fixed.
Entanglement and violation of Bell inequalities are aspects of quantum nonlocality that have been often confused in the past. It is now known that this equivalence is only true for pure states. Even though almost all the studies of quantum correlations at quantum phase transitions deal only with entanglement, we here argue that Bell inequalities can also reveal a general quantum phase transition. This is also shown for a particular case of two spin-12\frac{1}{2}21​ particles in an infinite one-dimensional chain described by the XXZ model. In this case, the Bell inequality is able to signal not only the first-order phase transition, but also the infinite-order Kosterlitz-Thouless quantum phase transition, which cannot be revealed either by the energy of the system nor by the bipartite entanglement. We also show that although the nearest-neighbor spins are entangled, they, unexpectedly, never violate the Bell inequality. This indicates that the type of entanglement which is relevant for quantum phase transition is not trivial, i.e., it cannot be revealed by the Bell inequality.
In quantum systems with infinitely many degrees of freedom, states can be infinitely entangled across a pair of subsystems. But are there different forms of infinite entanglement? To answer this, we provide a rigorous framework for studying information-theoretic properties in infinite systems, formulated in terms of von Neumann algebras. In this framework, we show that there are operational tasks that can distinguish between different forms of infinite entanglement. We find that the type classification of von Neumann algebras (types I, II, III, and their respective subtypes) is in one-to-one correspondence with operational entanglement properties. For instance, the Connes classification of type III factors corresponds to the system’s capability at the operational task of “embezzling” entanglement. Our findings promote the type classification from algebraic bookkeeping to a classification of infinite quantum systems based on their operational entanglement properties.
Quantum nonlocality in networks featuring multiple independent sources underpins large-scale quantum communication and poses fundamental challenges for its characterization. In this work, we construct a family of explicit nonlinear Bell inequalities to verify the nonlocality across the general multi-input quantum networks. The construction of these inequalities relies on the number of leaf nodes, a network parameter that can be identified by a linear-time algorithm. Our approach establishes a structural connection between bipartite full-correlation Bell inequalities and network Bell inequalities, enabling the analytical derivation of optimal quantum violations and the conditions under which they occur. We further quantify the upper bound on maximal violations achievable by arbitrary two-qubit mixed states in such networks, under separable measurements, and evaluate the noise robustness of the proposed inequalities via the visibilities of noisy states. Finally, we demonstrate that these inequalities can, in a device-independent manner, distinguish between network topologies of equal size that differ in the number of leaf nodes.