Quasi-Bell inequalities from symmetrized products of noncommuting qubit observables

Gamel, Omar E.; Fleming, Graham R.

doi:10.1063/1.4983918

Title: Quasi-Bell inequalities from symmetrized products of noncommuting qubit observables

Journal Article · Mon May 01 00:00:00 EDT 2017 · Journal of Mathematical Physics

DOI:https://doi.org/10.1063/1.4983918· OSTI ID:1393227

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Univ. of California, Berkeley, CA (United States). Dept. of Chemistry; Lawrence Berkeley National Lab. (LBNL), Berkeley, CA (United States). Molecular Biophysics and Integrated Bioimaging Division

Noncommuting observables cannot be simultaneously measured; however, under local hidden variable models, they must simultaneously hold premeasurement values, implying the existence of a joint probability distribution. We study the joint distributions of noncommuting observables on qubits, with possible criteria of positivity and the Fréchet bounds limiting the joint probabilities, concluding that the latter may be negative. We use symmetrization, justified heuristically and then more carefully via the Moyal characteristic function, to find the quantum operator corresponding to the product of noncommuting observables. This is then used to construct Quasi-Bell inequalities, Bell inequalities containing products of noncommuting observables, on two qubits. These inequalities place limits on the local hidden variable models that define joint probabilities for noncommuting observables. We also found that the Quasi-Bell inequalities have a quantum to classical violation as high as 3/2 on two qubit, higher than conventional Bell inequalities. Our result demonstrates the theoretical importance of noncommutativity in the nonlocality of quantum mechanics and provides an insightful generalization of Bell inequalities.

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Research Organization:: Lawrence Berkeley National Laboratory (LBNL), Berkeley, CA (United States)

Sponsoring Organization:: USDOE Office of Science (SC), Basic Energy Sciences (BES)

Grant/Contract Number:: AC02-05CH11231; AC03-76F000098

OSTI ID:: 1393227

Alternate ID(s):: OSTI ID: 1361904

Journal Information:: Journal of Mathematical Physics, Vol. 58, Issue 5; ISSN 0022-2488

Publisher:: American Institute of Physics (AIP)Copyright Statement

Country of Publication:: United States

Language:: English

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