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

Jan Hamhalter^a, Ekaterina A. Turilova<sup>b

a</sup> Czech Technical University in Prague, Faculty of Electrical Engineering, Technická 2, 166 27 Prague 6, Czech Republic
^b Kazan Federal University, Kremlevskaya ul. 18, Kazan, 420008 Russia

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.

Keywords: Bell's inequalities, JBW$^*$ algebra, JBW$^*$ triple.

UDC: 517.9

Received: June 15, 2023
Revised: July 19, 2023
Accepted: August 18, 2023

DOI: 10.4213/tm4367

 English version: Proceedings of the Steklov Institute of Mathematics, 2024, 324, 213–224

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