Quasi-Bell inequalities from symmetrized products of noncommuting qubit observables

doi:10.1063/1.4983918

Title: 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:

^[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:: 2017-05-01

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

DOE Contract Number:: AC02-05CH11231; AC03-76F000098

Resource Type:: Journal Article

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.

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

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                    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.

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@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},

  issn         = {0022-2488},

  number       = 5,

  volume       = 58,

  place        = {United States},

  year         = {2017},

  month        = {5}

}

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DOI: 10.1063/1.4983918

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Similar Records

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

Abstract

Citation Formats

Experimental Realization of Einstein-Podolsky-Rosen-Bohm Gedankenexperiment : A New Violation of Bell's Inequalities journal, July 1982

Noncommuting observables and local realism journal, November 2005

A complete set of multidimensional Bell inequalities journal, May 2012

Grothendieck’s constant and local models for noisy entangled quantum states journal, June 2006

Quantum analogues of the Bell inequalities. The case of two spatially separated domains journal, February 1987

Constantes de Grothendieck et fonctions de type positif sur les sphères journal, January 1979

Proposed Experiment to Test Local Hidden-Variable Theories journal, October 1969

Algorithmic Construction of Local Hidden Variable Models for Entangled Quantum States journal, November 2016

Local hidden–variable models for entangled quantum states journal, October 2014

Experimental Test of Bell's Inequalities Using Time- Varying Analyzers journal, December 1982

Unbounded Violations of Bipartite Bell Inequalities via Operator Space Theory journal, September 2010

Wringing out better Bell inequalities journal, August 1990

Quantum generalizations of Bell's inequality journal, March 1980

New Bell inequalities for the singlet state: Going beyond the Grothendieck bound journal, January 2008

On the Einstein Podolsky Rosen paradox journal, November 1964

Spin quasi-distribution functions journal, January 1994

A survey of the theory of characteristic functions journal, April 1972

Experimental Test of Local Hidden-Variable Theories journal, April 1972

Loophole-free Bell inequality violation using electron spins separated by 1.3 kilometres journal, October 2015

Experimental Tests of Realistic Local Theories via Bell's Theorem journal, August 1981

Simulating all Nonsignaling Correlations via Classical or Quantum Theory with Negative Probabilities journal, October 2013

The Statistical Interpretation of Quantum Mechanics journal, October 1970

All quantum observables in a hidden-variable model must commute simultaneously journal, February 2004

Quasi-probability distribution for spin-1/2 particles journal, July 1992

Joint probabilities of noncommuting operators and incompleteness of quantum mechanics journal, October 1988

A new interpretation of Bell's inequalities journal, July 1993

When is the wigner quasi-probability density non-negative? journal, October 1974

Hidden Variables, Joint Probability, and the Bell Inequalities journal, February 1982

More efficient Bell inequalities for Werner states journal, September 2008

A postulational framework for theories of simultaneous measurement of several observables journal, March 1973

Negativity and Contextuality are Equivalent Notions of Nonclassicality journal, July 2008

Généralisation du théorème des probabilités totales journal, January 1935

Joint distributions, quantum correlations, and commuting observables journal, July 1982

Two-party Bell inequalities derived from combinatorics via triangular elimination journal, November 2005

A survey of the theory of characteristic functions journal, April 1972

Entangled Bloch spheres: Bloch matrix and two-qubit state space journal, June 2016

Bell nonlocality journal, April 2014

Significant-Loophole-Free Test of Bell’s Theorem with Entangled Photons journal, December 2015

Strong Loophole-Free Test of Local Realism journal, December 2015

On the equivalence of nonlocality and nonpositivity of quasi-distributions in EPR correlations journal, May 1988

On the Formation of Quantum-Mechanical Operators journal, January 1959

On the physical interpretation of negative probabilities in Prugovecki's empirical theory of measurement journal, January 1997

On the Problem of Hidden Variables in Quantum Mechanics journal, July 1966

John Bell and the nature of the quantum world journal, October 2014

Correlation between Measurements in Quantum Theory journal, November 1961

Negative quasi-probability as a resource for quantum computation journal, November 2012

Quantum states with Einstein-Podolsky-Rosen correlations admitting a hidden-variable model journal, October 1989

Simultaneous Measurement of Noncommuting Observables journal, December 1966

Bell Inequalities for Arbitrarily High-Dimensional Systems journal, January 2002

Better local hidden variable models for two-qubit Werner states and an upper bound on the Grothendieck constant K G ( 3 ) journal, April 2017

Experimental Realization of Einstein-Podolsky-Rosen-Bohm Gedankenexperiment : A New Violation of Bell's Inequalities
journal, July 1982

Noncommuting observables and local realism
journal, November 2005

A complete set of multidimensional Bell inequalities
journal, May 2012

Grothendieck’s constant and local models for noisy entangled quantum states
journal, June 2006

Quantum analogues of the Bell inequalities. The case of two spatially separated domains
journal, February 1987

Constantes de Grothendieck et fonctions de type positif sur les sphères
journal, January 1979

Proposed Experiment to Test Local Hidden-Variable Theories
journal, October 1969

Algorithmic Construction of Local Hidden Variable Models for Entangled Quantum States
journal, November 2016

Local hidden–variable models for entangled quantum states
journal, October 2014

Experimental Test of Bell's Inequalities Using Time- Varying Analyzers
journal, December 1982

Unbounded Violations of Bipartite Bell Inequalities via Operator Space Theory
journal, September 2010

Wringing out better Bell inequalities
journal, August 1990

Quantum generalizations of Bell's inequality
journal, March 1980

New Bell inequalities for the singlet state: Going beyond the Grothendieck bound
journal, January 2008

On the Einstein Podolsky Rosen paradox
journal, November 1964

Spin quasi-distribution functions
journal, January 1994

A survey of the theory of characteristic functions
journal, April 1972

Experimental Test of Local Hidden-Variable Theories
journal, April 1972

Loophole-free Bell inequality violation using electron spins separated by 1.3 kilometres
journal, October 2015

Experimental Tests of Realistic Local Theories via Bell's Theorem
journal, August 1981

Simulating all Nonsignaling Correlations via Classical or Quantum Theory with Negative Probabilities
journal, October 2013

The Statistical Interpretation of Quantum Mechanics
journal, October 1970

All quantum observables in a hidden-variable model must commute simultaneously
journal, February 2004

Quasi-probability distribution for spin-1/2 particles
journal, July 1992

Joint probabilities of noncommuting operators and incompleteness of quantum mechanics
journal, October 1988

A new interpretation of Bell's inequalities
journal, July 1993

When is the wigner quasi-probability density non-negative?
journal, October 1974

Hidden Variables, Joint Probability, and the Bell Inequalities
journal, February 1982

More efficient Bell inequalities for Werner states
journal, September 2008

A postulational framework for theories of simultaneous measurement of several observables
journal, March 1973

Negativity and Contextuality are Equivalent Notions of Nonclassicality
journal, July 2008

Généralisation du théorème des probabilités totales
journal, January 1935

Joint distributions, quantum correlations, and commuting observables
journal, July 1982

Two-party Bell inequalities derived from combinatorics via triangular elimination
journal, November 2005

A survey of the theory of characteristic functions
journal, April 1972

Entangled Bloch spheres: Bloch matrix and two-qubit state space
journal, June 2016

Bell nonlocality
journal, April 2014

Significant-Loophole-Free Test of Bell’s Theorem with Entangled Photons
journal, December 2015

Strong Loophole-Free Test of Local Realism
journal, December 2015

On the equivalence of nonlocality and nonpositivity of quasi-distributions in EPR correlations
journal, May 1988

On the Formation of Quantum-Mechanical Operators
journal, January 1959

On the physical interpretation of negative probabilities in Prugovecki's empirical theory of measurement
journal, January 1997

On the Problem of Hidden Variables in Quantum Mechanics
journal, July 1966

John Bell and the nature of the quantum world
journal, October 2014

Correlation between Measurements in Quantum Theory
journal, November 1961

Negative quasi-probability as a resource for quantum computation
journal, November 2012

Quantum states with Einstein-Podolsky-Rosen correlations admitting a hidden-variable model
journal, October 1989

Simultaneous Measurement of Noncommuting Observables
journal, December 1966

Bell Inequalities for Arbitrarily High-Dimensional Systems
journal, January 2002

Better local hidden variable models for two-qubit Werner states and an upper bound on the Grothendieck constant $K_{G} (3)$
journal, April 2017