QuasiBell inequalities from symmetrized products of noncommuting qubit observables
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 QuasiBell 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 QuasiBell 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.
 Authors:

^{[1]}
;
^{[1]}
 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
 Publication Date:
 Grant/Contract Number:
 AC0205CH11231; AC0376F000098
 Type:
 Accepted Manuscript
 Journal Name:
 Journal of Mathematical Physics
 Additional Journal Information:
 Journal Volume: 58; Journal Issue: 5; Journal ID: ISSN 00222488
 Publisher:
 American Institute of Physics (AIP)
 Research Org:
 Lawrence Berkeley National Lab. (LBNL), Berkeley, CA (United States)
 Sponsoring Org:
 USDOE Office of Science (SC), Basic Energy Sciences (BES) (SC22)
 Country of Publication:
 United States
 Language:
 English
 Subject:
 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS
 OSTI Identifier:
 1393227
 Alternate Identifier(s):
 OSTI ID: 1361904
Gamel, Omar E., and Fleming, Graham R.. QuasiBell inequalities from symmetrized products of noncommuting qubit observables. United States: N. p.,
Web. doi:10.1063/1.4983918.
Gamel, Omar E., & Fleming, Graham R.. QuasiBell inequalities from symmetrized products of noncommuting qubit observables. United States. doi:10.1063/1.4983918.
Gamel, Omar E., and Fleming, Graham R.. 2017.
"QuasiBell inequalities from symmetrized products of noncommuting qubit observables". United States.
doi:10.1063/1.4983918. https://www.osti.gov/servlets/purl/1393227.
@article{osti_1393227,
title = {QuasiBell inequalities from symmetrized products of noncommuting qubit observables},
author = {Gamel, Omar E. and Fleming, Graham R.},
abstractNote = {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 QuasiBell 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 QuasiBell 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.},
doi = {10.1063/1.4983918},
journal = {Journal of Mathematical Physics},
number = 5,
volume = 58,
place = {United States},
year = {2017},
month = {5}
}