Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Inequality-type tests of commutativity of quantum observables using uncorrelated states

Koji Nagata

Published Aug 5, 2004• 0 citations
OpenAlex

Abstract

One of the essential features of quantum theory is the noncommutativity of observables. Moreover, under the non-contextual realistic interpretations of quantum theory, all quantum observables must commute {[J. D. Malley, Phys. Rev. A {\\bf 69}, 022118 (2004)]}. Experimental realizations of violations of Bell inequality say the noncommutativity of observables in an entangled state. Here we show that commutativity can be tested experimentally using uncorrelated states, with no dependence on the algebraic structure of the observables under study. The test is derived from the bounds of an inequality valid for all commuting observables. For $n$ spin-1/2 systems we show that an $n$-partite uncorrelated state violates the inequality by an amount that grows exponentially with $n$.

Related papers

Correlated observables in single-particle systems and field theoretic interpretations

Ian T. Durham

Bell-type experiments that test correlated observables typically involve measurements of spin or polarization on multi-particle systems in singlet states. These observables are all non-commuting and satisfy an uncertainty relation. Theoretically, the non-commuting nature should be independent of whether the singlet state consists of multiple particles or a single particle. Recent experiments in single neutron interferometry have in fact demonstrated this. In addition, if Bell-type inequalities can be found for experiments involving spin and polarization, the same should be true for experiments involving other non-commuting observables such as position and momentum as in the original EPR paper. As such, an experiment is proposed to measure (quantum mechanically) position and momentum for a single oscillator as a means for deriving a Bell-type inequality for these correlated observables. The experiment, if realizable, would shed light on the basic nature of matter, perhaps pointing to some form of self-entanglement, and would also help to further elucidate a possible mechanism behind the Heisenberg uncertainty principle. Violation of these inequalities would, in fact, offer yet another confirmation of the principle.

arXiv:quant-ph/0511048•Nov 6, 2005•0 citations

Inequalities for experimental tests of the Kochen-Specker theorem

Koji Nagata

We derive inequalities for n-partite states under the assumption that the hidden-variable theoretical joint probability distribution for any pair of commuting observables is equal to the quantum mechanical one. Fine showed that this assumption is connected to the no-hidden-variables theorem of Kochen and Specker (KS theorem). These inequalities give a way to experimentally test the KS theorem. The fidelity to the Bell states which is larger than 1∕2 is sufficient for the experimental confirmation of the KS theorem. Hence, the Werner state is enough to test experimentally the KS theorem. Furthermore, it is possible to test the KS theorem experimentally using uncorrelated states. An n-partite uncorrelated state violates the n-partite inequality derived here by an amount that grows exponentially with n.

arXiv:quant-ph/0312052•Journal of Mathematical Physics•Oct 1, 2005•8 citations

Correlated observables in single-particle systems

Ian T. Durham

Bell-type experiments that test correlated observables typically involve measurements of spin or polarization on multi-particle systems in singlet states. These observables are all non-commuting and satisfy an uncertainty relation. Theoretically, the non-commuting nature should be independent of whether the singlet state consists of multiple particles or a single particle. In addition, if Bell-type inequalities can be found for experiments involving spin and polarization, the same should be true for experiments involving other non-commuting observables such as position and momentum. In fact the original EPR paper used position and momentum, though in a different experimental situation. As such, an experiment is proposed to measure (quantum mechanically) position and momentum for a single oscillator as a means for deriving a Bell-type inequality for these correlated observables. The experiment, if realizable, would shed light on the basic nature of matter, perhaps pointing to some form of self-entanglement, and would also help to further elucidate a possible mechanism behind the Heisenberg uncertainty principle. Violation of these inequalities would, in fact, offer yet another confirmation of the principle.

•Nov 6, 2005•0 citations

Mermin's Bell inequality and operators' noncommutativity

Jose Luis Cereceda

The relationship between the noncommutativity of operators and the violation of the Bell inequality is exhibited in the light of the n-particle Bell-type inequality discovered by Mermin [PRL 65, 1838 (1990)]. It is shown that the maximal amount of violation of Mermin's inequality predicted by quantum mechanics decreases exponentially by a factor of 2^{-m/2} whenever any m among the n single-particle commutators happen to vanish.

•Jul 3, 2000•0 citations

Experimentally Testable State-Independent Violation of Bell-Type Inequalities for Quantum Contextuality

Adán Cabello

There are experimentally testable correlation Bell inequalities which are satisfied by any noncontextual theory but are violated by any quantum state. We discuss how to test one of them.

International Conference on Quantum, Nano and Micro Technologies (ICQNM)•Feb 1, 2009•0 citations

Experimentally Testable State-Independent Quantum Contextuality

Adán Cabello

We show that there are Bell-type inequalities for noncontextual theories that are violated by any quantum state. One of these inequalities between the correlations of compatible measurements is particularly suitable for testing this state-independent violation in an experiment.

arXiv:0808.2456•Physical Review Letters•Nov 19, 2008•407 citations
One of the essential features of quantum theory is the noncommutativity of observables. Moreover, under the non-contextual realistic interpretations of quantum theory, all quantum observables must commute {[J. D. Malley, Phys. Rev. A {\bf 69}, 022118 (2004)]}. Experimental realizations of violations of Bell inequality say the noncommutativity of observables in an entangled state. Here we show that commutativity can be tested experimentally using uncorrelated states, with no dependence on the algebraic structure of the observables under study. The test is derived from the bounds of an inequality valid for all commuting observables. For nnn spin-1/2 systems we show that an nnn-partite uncorrelated state violates the inequality by an amount that grows exponentially with nnn.
Bell-type experiments that test correlated observables typically involve measurements of spin or polarization on multi-particle systems in singlet states. These observables are all non-commuting and satisfy an uncertainty relation. Theoretically, the non-commuting nature should be independent of whether the singlet state consists of multiple particles or a single particle. Recent experiments in single neutron interferometry have in fact demonstrated this. In addition, if Bell-type inequalities can be found for experiments involving spin and polarization, the same should be true for experiments involving other non-commuting observables such as position and momentum as in the original EPR paper. As such, an experiment is proposed to measure (quantum mechanically) position and momentum for a single oscillator as a means for deriving a Bell-type inequality for these correlated observables. The experiment, if realizable, would shed light on the basic nature of matter, perhaps pointing to some form of self-entanglement, and would also help to further elucidate a possible mechanism behind the Heisenberg uncertainty principle. Violation of these inequalities would, in fact, offer yet another confirmation of the principle.
We derive inequalities for n-partite states under the assumption that the hidden-variable theoretical joint probability distribution for any pair of commuting observables is equal to the quantum mechanical one. Fine showed that this assumption is connected to the no-hidden-variables theorem of Kochen and Specker (KS theorem). These inequalities give a way to experimentally test the KS theorem. The fidelity to the Bell states which is larger than 1∕2 is sufficient for the experimental confirmation of the KS theorem. Hence, the Werner state is enough to test experimentally the KS theorem. Furthermore, it is possible to test the KS theorem experimentally using uncorrelated states. An n-partite uncorrelated state violates the n-partite inequality derived here by an amount that grows exponentially with n.
Bell-type experiments that test correlated observables typically involve measurements of spin or polarization on multi-particle systems in singlet states. These observables are all non-commuting and satisfy an uncertainty relation. Theoretically, the non-commuting nature should be independent of whether the singlet state consists of multiple particles or a single particle. In addition, if Bell-type inequalities can be found for experiments involving spin and polarization, the same should be true for experiments involving other non-commuting observables such as position and momentum. In fact the original EPR paper used position and momentum, though in a different experimental situation. As such, an experiment is proposed to measure (quantum mechanically) position and momentum for a single oscillator as a means for deriving a Bell-type inequality for these correlated observables. The experiment, if realizable, would shed light on the basic nature of matter, perhaps pointing to some form of self-entanglement, and would also help to further elucidate a possible mechanism behind the Heisenberg uncertainty principle. Violation of these inequalities would, in fact, offer yet another confirmation of the principle.