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Proceedings of the Steklov Institute of Mathematics

Violation of Bell’s Inequalities in Jordan Triples and Jordan Algebras

Jan Hamhalter, Ekaterina Aleksandrovna Turilova

Published Mar 1, 2024• 0 citations
Publisher

Abstract

We formulate and prove Bell’s inequalities in the realm of JB $$^*$$ triples and JB $$^*$$ algebras. We show that the maximal violation of Bell’s inequalities occurs in any JBW $$^*$$ triple containing a nonassociative $$2$$ -Peirce subspace. Moreover, we show that the violation of Bell’s inequalities in a nonmodular JBW $$^*$$ algebra and in an essentially nonmodular JBW $$^*$$ triple is generic. We describe the structure of maximal violators and its relation to the spin factor. In addition, we present a synthesis of available results based on a unified geometric approach.

DOI
10.1134/s008154382401019x

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We formulate and prove Bell’s inequalities in the realm of JB ∗^*∗ triples and JB ∗^*∗ algebras. We show that the maximal violation of Bell’s inequalities occurs in any JBW ∗^*∗ triple containing a nonassociative 222 -Peirce subspace. Moreover, we show that the violation of Bell’s inequalities in a nonmodular JBW ∗^*∗ algebra and in an essentially nonmodular JBW ∗^*∗ triple is generic. We describe the structure of maximal violators and its relation to the spin factor. In addition, we present a synthesis of available results based on a unified geometric approach.
The paper deals with the structure of Bell’s inequalities (Clauser, Horne, Shimony, and Holt (CHSH) form). It is established that 2 is the universal bound for Bell’s correlations given by a general correlation duality on a complex linear space. Using geometric arguments the maximal violation of Bell’s inequalities is described on this abstract level. It is proved that Bell’s inequalities are maximally violated for general ∗-algebras and a faithful state exactly when the corresponding elements are the Pauli spin matrices. Interesting structural consequences of this result are derived. They demonstrate that the maximal violation requires a very “nonclassical” position of the local systems. Moreover, the maximal strength of Bell’s correlations for a general state with respect to nearly commuting subalgebras is characterized in terms of the spin systems and tracial properties of the local states.
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In this paper, it is proved that the maximal violation of Mermin's inequalities of nnn qubits occurs only for GHZ's states and the states obtained from them by local unitary transformations. The key point of our argument involved here is by using the certain algebraic properties that Pauli's matrices satisfy, which is based on the determination of local spin observables of the associated Bell-Mermin operators.
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
We discuss general Bell inequalities for bipartite and multipartite systems, emphasizing the connection with convex geometry on the mathematical side, and the communication aspects on the physical side. Known results on families of generalized Bell inequalities are summarized. We investigate maximal violations of Bell inequalities as well as states not violating (certain) Bell inequalities. Finally, we discuss the relation between Bell inequality violations and entanglement properties currently discussed in quantum information theory.