# Quasi-Bell inequalities from symmetrized products of noncommuting qubit observables

## Abstract

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.

- Authors:

- 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:

- Research Org.:
- Lawrence Berkeley National Lab. (LBNL), Berkeley, CA (United States)

- Sponsoring Org.:
- USDOE Office of Science (SC), Basic Energy Sciences (BES) (SC-22)

- OSTI Identifier:
- 1393227

- Alternate Identifier(s):
- OSTI ID: 1361904

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

- Resource Type:
- Journal Article: Accepted Manuscript

- Journal Name:
- Journal of Mathematical Physics

- Additional Journal Information:
- Journal Volume: 58; Journal Issue: 5; Journal ID: ISSN 0022-2488

- Publisher:
- American Institute of Physics (AIP)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS

### Citation Formats

```
Gamel, Omar E., and Fleming, Graham R..
```*Quasi-Bell inequalities from symmetrized products of noncommuting qubit observables*. United States: N. p., 2017.
Web. doi:10.1063/1.4983918.

```
Gamel, Omar E., & Fleming, Graham R..
```*Quasi-Bell inequalities from symmetrized products of noncommuting qubit observables*. United States. doi:10.1063/1.4983918.

```
Gamel, Omar E., and Fleming, Graham R.. Mon .
"Quasi-Bell 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 = {Quasi-Bell 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 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.},

doi = {10.1063/1.4983918},

journal = {Journal of Mathematical Physics},

number = 5,

volume = 58,

place = {United States},

year = {Mon May 01 00:00:00 EDT 2017},

month = {Mon May 01 00:00:00 EDT 2017}

}