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
- 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.
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},
issn = {0022-2488},
number = 5,
volume = 58,
place = {United States},
year = {2017},
month = {5}
}
Works referenced in this record:
Experimental Realization of Einstein-Podolsky-Rosen-Bohm Gedankenexperiment : A New Violation of Bell's Inequalities
journal, July 1982
- Aspect, Alain; Grangier, Philippe; Roger, Gérard
- Physical Review Letters, Vol. 49, Issue 2
Noncommuting observables and local realism
journal, November 2005
- Malley, James D.; Fine, Arthur
- Physics Letters A, Vol. 347, Issue 1-3
A complete set of multidimensional Bell inequalities
journal, May 2012
- Arnault, François
- Journal of Physics A: Mathematical and Theoretical, Vol. 45, Issue 25
Grothendieck’s constant and local models for noisy entangled quantum states
journal, June 2006
- Acín, Antonio; Gisin, Nicolas; Toner, Benjamin
- Physical Review A, Vol. 73, Issue 6
Quantum analogues of the Bell inequalities. The case of two spatially separated domains
journal, February 1987
- Tsirel'son, B. S.
- Journal of Soviet Mathematics, Vol. 36, Issue 4
Constantes de Grothendieck et fonctions de type positif sur les sphères
journal, January 1979
- Krivine, J. L.
- Advances in Mathematics, Vol. 31, Issue 1
Proposed Experiment to Test Local Hidden-Variable Theories
journal, October 1969
- Clauser, John F.; Horne, Michael A.; Shimony, Abner
- Physical Review Letters, Vol. 23, Issue 15
Algorithmic Construction of Local Hidden Variable Models for Entangled Quantum States
journal, November 2016
- Hirsch, Flavien; Quintino, Marco Túlio; Vértesi, Tamás
- Physical Review Letters, Vol. 117, Issue 19
Local hidden–variable models for entangled quantum states
journal, October 2014
- Augusiak, R.; Demianowicz, M.; Acín, A.
- Journal of Physics A: Mathematical and Theoretical, Vol. 47, Issue 42
Experimental Test of Bell's Inequalities Using Time- Varying Analyzers
journal, December 1982
- Aspect, Alain; Dalibard, Jean; Roger, Gérard
- Physical Review Letters, Vol. 49, Issue 25
Unbounded Violations of Bipartite Bell Inequalities via Operator Space Theory
journal, September 2010
- Junge, M.; Palazuelos, C.; Pérez-García, D.
- Communications in Mathematical Physics, Vol. 300, Issue 3
Wringing out better Bell inequalities
journal, August 1990
- Braunstein, Samuel L.; Caves, Carlton M.
- Annals of Physics, Vol. 202, Issue 1
Quantum generalizations of Bell's inequality
journal, March 1980
- Cirel'son, B. S.
- Letters in Mathematical Physics, Vol. 4, Issue 2
New Bell inequalities for the singlet state: Going beyond the Grothendieck bound
journal, January 2008
- Pitowsky, Itamar
- Journal of Mathematical Physics, Vol. 49, Issue 1
On the Einstein Podolsky Rosen paradox
journal, November 1964
- Bell, J. S.
- Physics Physique Fizika, Vol. 1, Issue 3
Spin quasi-distribution functions
journal, January 1994
- Scully, M. O.; Wódkiewicz, K.
- Foundations of Physics, Vol. 24, Issue 1
A survey of the theory of characteristic functions
journal, April 1972
- Lukacs, Eugene
- Advances in Applied Probability, Vol. 4, Issue 1
Experimental Test of Local Hidden-Variable Theories
journal, April 1972
- Freedman, Stuart J.; Clauser, John F.
- Physical Review Letters, Vol. 28, Issue 14
Loophole-free Bell inequality violation using electron spins separated by 1.3 kilometres
journal, October 2015
- Hensen, B.; Bernien, H.; Dréau, A. E.
- Nature, Vol. 526, Issue 7575
Experimental Tests of Realistic Local Theories via Bell's Theorem
journal, August 1981
- Aspect, Alain; Grangier, Philippe; Roger, Gérard
- Physical Review Letters, Vol. 47, Issue 7
Simulating all Nonsignaling Correlations via Classical or Quantum Theory with Negative Probabilities
journal, October 2013
- Al-Safi, Sabri W.; Short, Anthony J.
- Physical Review Letters, Vol. 111, Issue 17
The Statistical Interpretation of Quantum Mechanics
journal, October 1970
- Ballentine, L. E.
- Reviews of Modern Physics, Vol. 42, Issue 4
All quantum observables in a hidden-variable model must commute simultaneously
journal, February 2004
- Malley, James D.
- Physical Review A, Vol. 69, Issue 2
Quasi-probability distribution for spin-1/2 particles
journal, July 1992
- Chandler, C.; Cohen, L.; Lee, C.
- Foundations of Physics, Vol. 22, Issue 7
Joint probabilities of noncommuting operators and incompleteness of quantum mechanics
journal, October 1988
- Barut, A. O.; Božić, M.; Marić, Z.
- Foundations of Physics, Vol. 18, Issue 10
A new interpretation of Bell's inequalities
journal, July 1993
- Sudarshan, E. C. G.; Rothman, Tony
- International Journal of Theoretical Physics, Vol. 32, Issue 7
When is the wigner quasi-probability density non-negative?
journal, October 1974
- Hudson, R. L.
- Reports on Mathematical Physics, Vol. 6, Issue 2
Hidden Variables, Joint Probability, and the Bell Inequalities
journal, February 1982
- Fine, Arthur
- Physical Review Letters, Vol. 48, Issue 5
More efficient Bell inequalities for Werner states
journal, September 2008
- Vértesi, T.
- Physical Review A, Vol. 78, Issue 3
A postulational framework for theories of simultaneous measurement of several observables
journal, March 1973
- Prugovečki, Eduard
- Foundations of Physics, Vol. 3, Issue 1
Negativity and Contextuality are Equivalent Notions of Nonclassicality
journal, July 2008
- Spekkens, Robert W.
- Physical Review Letters, Vol. 101, Issue 2
Généralisation du théorème des probabilités totales
journal, January 1935
- Fréchet, Maurice
- Fundamenta Mathematicae, Vol. 25
Joint distributions, quantum correlations, and commuting observables
journal, July 1982
- Fine, Arthur
- Journal of Mathematical Physics, Vol. 23, Issue 7
Two-party Bell inequalities derived from combinatorics via triangular elimination
journal, November 2005
- Avis, David; Imai, Hiroshi; Ito, Tsuyoshi
- Journal of Physics A: Mathematical and General, Vol. 38, Issue 50
A survey of the theory of characteristic functions
journal, April 1972
- Lukacs, Eugene
- Advances in Applied Probability, Vol. 4, Issue 01
Entangled Bloch spheres: Bloch matrix and two-qubit state space
journal, June 2016
- Gamel, Omar
- Physical Review A, Vol. 93, Issue 6
Bell nonlocality
journal, April 2014
- Brunner, Nicolas; Cavalcanti, Daniel; Pironio, Stefano
- Reviews of Modern Physics, Vol. 86, Issue 2
Significant-Loophole-Free Test of Bell’s Theorem with Entangled Photons
journal, December 2015
- Giustina, Marissa; Versteegh, Marijn A. M.; Wengerowsky, Sören
- Physical Review Letters, Vol. 115, Issue 25
Strong Loophole-Free Test of Local Realism
journal, December 2015
- Shalm, Lynden K.; Meyer-Scott, Evan; Christensen, Bradley G.
- Physical Review Letters, Vol. 115, Issue 25
On the equivalence of nonlocality and nonpositivity of quasi-distributions in EPR correlations
journal, May 1988
- Wódkiewicz, K.
- Physics Letters A, Vol. 129, Issue 1
On the Formation of Quantum-Mechanical Operators
journal, January 1959
- Shewell, John Robert
- American Journal of Physics, Vol. 27, Issue 1
On the physical interpretation of negative probabilities in Prugovecki's empirical theory of measurement
journal, January 1997
- Khrennikov, A.
- Canadian Journal of Physics, Vol. 75, Issue 5
On the Problem of Hidden Variables in Quantum Mechanics
journal, July 1966
- Bell, John S.
- Reviews of Modern Physics, Vol. 38, Issue 3
John Bell and the nature of the quantum world
journal, October 2014
- Bertlmann, Reinhold A.
- Journal of Physics A: Mathematical and Theoretical, Vol. 47, Issue 42
Correlation between Measurements in Quantum Theory
journal, November 1961
- Margenau, Henry; Hill, Robert Nyden
- Progress of Theoretical Physics, Vol. 26, Issue 5
Negative quasi-probability as a resource for quantum computation
journal, November 2012
- Veitch, Victor; Ferrie, Christopher; Gross, David
- New Journal of Physics, Vol. 14, Issue 11
Quantum states with Einstein-Podolsky-Rosen correlations admitting a hidden-variable model
journal, October 1989
- Werner, Reinhard F.
- Physical Review A, Vol. 40, Issue 8
Simultaneous Measurement of Noncommuting Observables
journal, December 1966
- She, C. Y.; Heffner, H.
- Physical Review, Vol. 152, Issue 4
Bell Inequalities for Arbitrarily High-Dimensional Systems
journal, January 2002
- Collins, Daniel; Gisin, Nicolas; Linden, Noah
- Physical Review Letters, Vol. 88, Issue 4
Better local hidden variable models for two-qubit Werner states and an upper bound on the Grothendieck constant
journal, April 2017
- Hirsch, Flavien; Quintino, Marco Túlio; Vértesi, Tamás
- Quantum, Vol. 1