Abstract
The structure of maximal violators of Bell’s inequalities for Jordan algebras is investigated. It is proved that the spin factor V 2 is responsible for maximal values of Bell’s correlations in a faithful state. In this situation maximally correlated subsystems must overlap in a nonassociative subalgebra. For operator commuting subalgebras it is shown that maximal violators have the structure of the spin systems and that the global state (faithful on local subalgebras) acts as the trace on local subalgebras.
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References
Bell, J.: On the Einstein-Podolsky-Rosen paradox. Physics 1, 195–200 (1964)
Bohata, M., Hamhalter, J.: Maximal violation of Bell’s inequalities and Pauli spin matrices. J. Math. Phys. 50, 082101 (2009)
Emch, G.: Algebraic Methods in Statistical Mechanics and Quantum Filed Theory. Wiley, New York (1972)
Hamhalter, J.: Quantum Measure Theory. Fundamental Theories of Physics. Kluwer Academic, Dordrecht/Boston/London (2003)
Hanche-Olsen, H., Størmer, E.: Jordan Operator Algebras. Pitman, Boston/London/Melbourne (1984)
Nielsen, M.A., Chuang, I.L.: Quantum Computation and Quantum Information. Cambridge University Press, Cambridge (2000)
Pedersen, G.K., Størmer, E.: Traces on Jordan algebras. Canad. J. Math. 34(2), 370–373 (1982)
Rédei, M.: Quantum Logic in Algebraic Approach. Fundamental Theories of Physics. Kluwer Academic, Dordrecht/Boston/London (1998)
Summers, S.J.: On the independence of local algebras in quantum field theory. Rev. Math. Phys. 2(2), 201–247 (1990)
Summers, S.J.: Bell’s inequalities and algebraic structure. In: Doplicher, S., Longo, R., Roberts, J.E., Zsido, L. (eds.) Operator Algebras and Quantum Field Theory, pp. 633–646. International Press (1997)
Summers, S.J., Werner, R.F.: Bell’s inequalities and quantum field theory I. General setting. J. Math. Phys. 28, 2440–2447 (1987)
Summers, S.J., Werner, R.F.: Bell’s inequalities and quantum field theory. II. Bell’s inequalities are maximally violated in the vacuum. J. Math. Phys. 28, 2448–2456 (1987)
Summers, S.J., Werner, R.F.: Maximal violation of Bell’s inequalities is generic in quantum field theory. Commun. Math. Phys. 110, 247–259 (1987)
Summers, S.J., Werner, R.F.: Maximal violation of Bell’s inequalities for algebras of observables in tangent spacetime regions. Ann. Inst. H. Poincaré 49, 215–243 (1988)
Summers, S.J., Werner, R.F.: On Bell’s inequalities and algebraic invariants. Lett. Math. Phys. 33, 321–334 (1995)
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Bohata, M., Hamhalter, J. Bell’s Correlations and Spin Systems. Found Phys 40, 1065–1075 (2010). https://doi.org/10.1007/s10701-009-9401-y
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DOI: https://doi.org/10.1007/s10701-009-9401-y