arXiv:0803.0913 · Physical Review A

Quaternions, octonions, and Bell-type inequalities

Evgeny V Shchukin, W. Vogel

Published Sep 4, 2008• 11 citations

Abstract

Multipartite Bell-type inequalities are derived for general systems. They involve up to eight observables with arbitrary spectra on each site. These inequalities are closely related to the algebras of quaternions and octonions.

DOI: 10.1103/physreva.78.032104

Related papers

All tight multipartite Bell correlation inequalities for three dichotomic observables per observer

Marek Żukowski

A derivation of the full set of Bell inequalities involving correlation functions, for two parties, with binary observables, and three possible local settings. The procedure can be extended straightforwardly to multiparty correlations.

arXiv:quant-ph/0611086Nov 8, 20064 citations

Bell's Inequalities and Algebraic Structure

Stephen J. Summers

We provide an overview of the connections between Bell's inequalities and algebraic structure.

arXiv:funct-an/9701003Jan 10, 199715 citations

Violations of multisetting quaternion and octonion Bell inequalities

Simon Kiesewetter, P. D. Drummond, M. D. Reid

Physical Review ASep 16, 20151 citation

Multipartite Bell-type inequalities for arbitrary numbers of settings and outcomes per site

Elena R. Loubenets

arXiv:0804.4046Journal of Physics A Mathematical and TheoreticalOct 7, 200822 citations

Bell inequalities for general quantum observables

Karl-Peter Marzlin, T. A. Osborn

Bulletin of the American Physical SocietyJun 7, 20130 citations

Compact Bell inequalities for multipartite experiments

Yu-Chun Wu, Marek Żukowski, Jing‐Ling Chen +1 more

arXiv:1302.6698Physical Review AAug 29, 201311 citations

arXiv:0803.0913 · Physical Review A

Quaternions, octonions, and Bell-type inequalities

Evgeny V Shchukin, W. Vogel

Published Sep 4, 2008• 11 citations

arXiv HTML PDF

Abstract

DOI: 10.1103/physreva.78.032104

Related papers

All tight multipartite Bell correlation inequalities for three dichotomic observables per observer

Marek Żukowski

arXiv:quant-ph/0611086Nov 8, 20064 citations

Bell's Inequalities and Algebraic Structure

Stephen J. Summers

We provide an overview of the connections between Bell's inequalities and algebraic structure.

arXiv:funct-an/9701003Jan 10, 199715 citations

Violations of multisetting quaternion and octonion Bell inequalities

Simon Kiesewetter, P. D. Drummond, M. D. Reid

We examine the $N$ -partite quaternion and octonion dichotomic Bell inequalities derived by Vogel and Shchukin utilizing the square identities of Euler and Degen, which apply to experiments with $M$ settings at each spatially separated site. We reveal these inequalities to be violated by the Greenberger-Horne-Zeilinger state, for $N\ensuremath{\ge}3$ and $2\ensuremath{\le}M\ensuremath{\le}8$. Violations are robust with respect to loss, the threshold detection efficiency being $\ensuremath{\eta}>{2}^{\frac{2}{N}\ensuremath{-}1}$ for all $M$ , implying violations for efficiencies as low as $\ensuremath{\eta}\ensuremath{\sim}50%$ at each site, as $N\ensuremath{\rightarrow}\ensuremath{\infty}$.We examine the $N$-partite quaternion and octonion dichotomic Bell inequalities derived by Vogel and Shchukin utilizing the square identities of Euler and Degen, which apply to experiments with $M$ settings at each spatially separated site. We reveal these inequalities to be violated by the Greenberger-Horne-Zeilinger state, for $N\ensuremath{\ge}3$ and $2\ensuremath{\le}M\ensuremath{\le}8$. Violations are robust with respect to loss, the threshold detection efficiency being $\ensuremath{\eta}>{2}^{\frac{2}{N}\ensuremath{-}1}$ for all $M$, implying violations for efficiencies as low as $\ensuremath{\eta}\ensuremath{\sim}50%$ at each site, as $N\ensuremath{\rightarrow}\ensuremath{\infty}$.

Physical Review ASep 16, 20151 citation

Multipartite Bell-type inequalities for arbitrary numbers of settings and outcomes per site

Elena R. Loubenets

We introduce a single general representation incorporating in a unique manner all Bell-type inequalities for a multipartite correlation scenario with an arbitrary number of settings and any spectral type of outcomes at each site. Specifying this general representation for correlation functions, we prove that the form of any correlation Bell-type inequality does not depend on spectral types of outcomes, in particular, on their numbers at different sites, and is determined only by extremal values of outcomes at each site. We also specify the general form of bounds in Bell-type inequalities on joint probabilities. Our approach to the derivation of Bell-type inequalities is universal, concise and can be applied to a multipartite correlation experiment with outcomes of any spectral type, discrete or continuous. We, in particular, prove that, for an N -partite quantum state, possibly, infinite dimensional, admitting the -setting LHV description, the Mermin–Klyshko inequality holds for any two bounded quantum observables per site, not necessarily dichotomic.We introduce a single general representation incorporating in a unique manner all Bell-type inequalities for a multipartite correlation scenario with an arbitrary number of settings and any spectral type of outcomes at each site. Specifying this general representation for correlation functions, we prove that the form of any correlation Bell-type inequality does not depend on spectral types of outcomes, in particular, on their numbers at different sites, and is determined only by extremal values of outcomes at each site. We also specify the general form of bounds in Bell-type inequalities on joint probabilities. Our approach to the derivation of Bell-type inequalities is universal, concise and can be applied to a multipartite correlation experiment with outcomes of any spectral type, discrete or continuous. We, in particular, prove that, for an N -partite quantum state, possibly, infinite dimensional, admitting the -setting LHV description, the Mermin–Klyshko inequality holds for any two bounded quantum observables per site, not necessarily dichotomic.

arXiv:0804.4046Journal of Physics A Mathematical and TheoreticalOct 7, 200822 citations

Bell inequalities for general quantum observables

Karl-Peter Marzlin, T. A. Osborn

Bulletin of the American Physical SocietyJun 7, 20130 citations

Compact Bell inequalities for multipartite experiments

Yu-Chun Wu, Marek Żukowski, Jing‐Ling Chen +1 more

A method for construction of multipartite Clauser-Horne-Shimony-Holt- (CHSH-) type Bell inequalities, for the case of local binary observables, is presented. The standard CHSH-type Bell inequalities can be obtained as special cases. An iterative method for deriving $(N+1)$ -partite CHSH-type Bell inequalities from $N$ -partite ones is derived. As an application, compact Bell inequalities for eight observers involving just four correlation functions are proposed. They require much less experimental effort than standard methods and thus are experimentally friendly in multiphoton experiments.A method for construction of multipartite Clauser-Horne-Shimony-Holt- (CHSH-) type Bell inequalities, for the case of local binary observables, is presented. The standard CHSH-type Bell inequalities can be obtained as special cases. An iterative method for deriving $(N+1)$-partite CHSH-type Bell inequalities from $N$-partite ones is derived. As an application, compact Bell inequalities for eight observers involving just four correlation functions are proposed. They require much less experimental effort than standard methods and thus are experimentally friendly in multiphoton experiments.

arXiv:1302.6698Physical Review AAug 29, 201311 citations

We examine the

N

-partite quaternion and octonion dichotomic Bell inequalities derived by Vogel and Shchukin utilizing the square identities of Euler and Degen, which apply to experiments with

M

settings at each spatially separated site. We reveal these inequalities to be violated by the Greenberger-Horne-Zeilinger state, for $N\ensuremath{\ge}3$ and $2\ensuremath{\le}M\ensuremath{\le}8$. Violations are robust with respect to loss, the threshold detection efficiency being $\ensuremath{\eta}>{2}^{\frac{2}{N}\ensuremath{-}1}$ for all

M

, implying violations for efficiencies as low as $\ensuremath{\eta}\ensuremath{\sim}50%$ at each site, as $N\ensuremath{\rightarrow}\ensuremath{\infty}$.

(N+1)

-partite CHSH-type Bell inequalities from

N

-partite ones is derived. As an application, compact Bell inequalities for eight observers involving just four correlation functions are proposed. They require much less experimental effort than standard methods and thus are experimentally friendly in multiphoton experiments.