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

Physical Review A

Violations of multisetting quaternion and octonion Bell inequalities

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

Published Sep 16, 2015• 1 citation
Publisher

Abstract

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 $N\ensuremath{\rightarrow}\ensuremath{\infty}$.

DOI
10.1103/physreva.92.032118

Related papers

Two-setting multisite Bell inequalities for loophole-free tests with up to 50 % loss

M. D. Reid

We consider Bell experiments with $N$ spatially separated qubits where loss is present and restrict ourselves to two measurement settings per site. We note the Mermin-Ardehali-Belinskii-Klyshko (MABK) Bell inequalities do not present a tight bound for the predictions of local hidden variable (LHV) theories. The Holder-type Bell inequality derived by Cavalcanti, Foster, Reid, and Drummond [E. G. Cavalcanti, C. J. Foster, M. D. Reid, and P. D. Drummond, Phys. Rev. Lett. 99, 210405 (2007)] provides a tighter bound, for high losses. We analyze the actual tight bound for the MABK inequalities, given the measure $W={\ensuremath{\prod}}_{k=1}^{N}{\ensuremath{\eta}}_{k}$ of overall detection efficiency, where ${\ensuremath{\eta}}_{k}$ is the efficiency at site $k$. Using these inequalities, we confirm that the maximally entangled Greenberger-Horne-Zeilinger state enables loophole-free falsification of LHV theories provided ${\ensuremath{\prod}}_{k=1}^{N}{\ensuremath{\eta}}_{k}>{2}^{(2\ensuremath{-}N)}$, which implies a symmetric threshold efficiency of $\ensuremath{\eta}\ensuremath{\rightarrow}50%$, as $N\ensuremath{\rightarrow}\ensuremath{\infty}$. Furthermore, loophole-free violations remain possible, even when the efficiency at some sites is reduced well below $0.5$, provided $N>3$.

arXiv:1310.2689•Physical Review A•Jun 18, 2013•2 citations

Variants of Bell inequalities

Zeqian Chen

A family of Bell-type inequalities is present, which are constructed directly from the "standard" Bell inequalities involving two dichotomic observables per site. It is shown that the inequalities are violated by all the generalized Greenberger-Horne-Zeilinger states of multiqubits. Remarkably, our new inequalities can provide stronger non-locality tests in a sense that the local reality inequalities are exponentially stronger than the corresponding multipartite separability inequalities. This reveals that the exponential violation of local realism by separable states is an interesting consequence of quantum fluctuation of multipartite systems.

arXiv:quant-ph/0611126•Nov 13, 2006•0 citations

Bell inequalities tailored to the Greenberger-Horne-Zeilinger states of arbitrary local dimension

Remigiusz Augusiak, Alexia Salavrakos, Tura i Brugués, J. +1 more

Abstract In device-independent quantum information processing Bell inequalities are not only used as detectors of nonlocality, but also as certificates of relevant quantum properties. In order for these certificates to work, one very often needs Bell inequalities that are maximally violated by specific quantum states. Recently, in Salavrakos et al (2017 Phys. Rev. Lett. 119 040402) a general class of Bell inequalities, with arbitrary numbers of measurements and outcomes, has been designed, which are maximally violated by the maximally entangled states of two quantum systems of arbitrary dimension. In this work, we generalize these results to the multipartite scenario and obtain a general class of Bell inequalities maximally violated by the Greenberger–Horne–Zeilinger states of any number of parties and any local dimension. We then derive analytically their maximal quantum and nonsignaling values. We also obtain analytically the bound for detecting genuine nonlocality and compute the fully local bound for a few exemplary cases. Moreover, we consider the question of adapting this class of inequalities to partially entangled Greenberger–Horne–Zeilinger-like states for some special cases of low dimension and small number of parties. Through numerical methods, we find classes of inequalities maximally violated by these partially entangled states.

•Nov 1, 2019•15 citations

Nonclassicality threshold for the three-qubit Greenberger-Horne-Zeilinger state

Tamás Vértesi, Károly F. Pál

The nonclassical properties of the noisy three-qubit Greenberger-Horne-Zeilinger (GHZ) states, ${\ensuremath{\rho}}_{v}=v|\text{GHZ}\ensuremath{\rangle}\ensuremath{\langle}\text{GHZ}|+(1\ensuremath{-}v)\mathbb{1}/8$ parameterized by the visibility $0\ensuremath{\le}v\ensuremath{\le}1$ are investigated. Based on the violation of the $2\ifmmode\times\else\texttimes\fi{}2\ifmmode\times\else\texttimes\fi{}2$-setting Mermin inequality, ${\ensuremath{\rho}}_{v}$ is nonclassical for the parameter range $1/2<v\ensuremath{\le}1$. It has been posed whether additional settings would allow to lower the threshold visibility. Here we report on Bell inequalities giving a threshold value smaller than $v=1/2$. This rules out the possibility of a local hidden variable model in the limit of $v=1/2$. In particular, the lowest threshold visibility we found is $v=0.496\phantom{\rule{0.16em}{0ex}}057$, attainable with $5\ifmmode\times\else\texttimes\fi{}5\ifmmode\times\else\texttimes\fi{}5$ settings, whereas the most economical one in number of settings corresponds to $3\ifmmode\times\else\texttimes\fi{}3\ifmmode\times\else\texttimes\fi{}4$ settings. The method which enabled us to obtain these results, and in particular the about 10 000 tight Bell inequalities giving $v<1/2$, are also discussed in detail.

arXiv:1108.1998•Physical Review A•Oct 28, 2011•10 citations

Bell inequalities tailored to the Greenberger–Horne–Zeilinger states of arbitrary local dimension

R Augusiak, Alexia Salavrakos, J Tura +1 more

Abstract In device-independent quantum information processing Bell inequalities are not only used as detectors of nonlocality, but also as certificates of relevant quantum properties. In order for these certificates to work, one very often needs Bell inequalities that are maximally violated by specific quantum states. Recently, in Salavrakos et al (2017 Phys. Rev. Lett. 119 040402) a general class of Bell inequalities, with arbitrary numbers of measurements and outcomes, has been designed, which are maximally violated by the maximally entangled states of two quantum systems of arbitrary dimension. In this work, we generalize these results to the multipartite scenario and obtain a general class of Bell inequalities maximally violated by the Greenberger–Horne–Zeilinger states of any number of parties and any local dimension. We then derive analytically their maximal quantum and nonsignaling values. We also obtain analytically the bound for detecting genuine nonlocality and compute the fully local bound for a few exemplary cases. Moreover, we consider the question of adapting this class of inequalities to partially entangled Greenberger–Horne–Zeilinger-like states for some special cases of low dimension and small number of parties. Through numerical methods, we find classes of inequalities maximally violated by these partially entangled states.

arXiv:1907.10116•New Journal of Physics•Oct 14, 2019•16 citations

Quaternions, octonions, and Bell-type inequalities

Evgeny V Shchukin, W. Vogel

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.

arXiv:0803.0913•Physical Review A•Sep 4, 2008•11 citations
We examine the NNN-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 MMM settings at each spatially separated site. We reveal these inequalities to be violated by the Greenberger-Horne-Zeilinger state, for N\ensuremath≥3N\ensuremath{\ge}3N\ensuremath≥3 and 2\ensuremath≤M\ensuremath≤82\ensuremath{\le}M\ensuremath{\le}82\ensuremath≤M\ensuremath≤8. Violations are robust with respect to loss, the threshold detection efficiency being \ensuremathη>22N\ensuremath−1\ensuremath{\eta}>{2}^{\frac{2}{N}\ensuremath{-}1}\ensuremathη>2N2​\ensuremath−1 for all MMM, implying violations for efficiencies as low as \ensuremathη\ensuremath∼50\ensuremath{\eta}\ensuremath{\sim}50%\ensuremathη\ensuremath∼50 at each site, as N\ensuremath→\ensuremath∞N\ensuremath{\rightarrow}\ensuremath{\infty}N\ensuremath→\ensuremath∞.
We consider Bell experiments with NNN spatially separated qubits where loss is present and restrict ourselves to two measurement settings per site. We note the Mermin-Ardehali-Belinskii-Klyshko (MABK) Bell inequalities do not present a tight bound for the predictions of local hidden variable (LHV) theories. The Holder-type Bell inequality derived by Cavalcanti, Foster, Reid, and Drummond [E. G. Cavalcanti, C. J. Foster, M. D. Reid, and P. D. Drummond, Phys. Rev. Lett. 99, 210405 (2007)] provides a tighter bound, for high losses. We analyze the actual tight bound for the MABK inequalities, given the measure $W={\ensuremath{\prod}}_{k=1}^{N}{\ensuremath{\eta}}_{k}$ of overall detection efficiency, where ${\ensuremath{\eta}}_{k}$ is the efficiency at site kkk. Using these inequalities, we confirm that the maximally entangled Greenberger-Horne-Zeilinger state enables loophole-free falsification of LHV theories provided ${\ensuremath{\prod}}_{k=1}^{N}{\ensuremath{\eta}}_{k}>{2}^{(2\ensuremath{-}N)}$, which implies a symmetric threshold efficiency of $\ensuremath{\eta}\ensuremath{\rightarrow}50%$, as $N\ensuremath{\rightarrow}\ensuremath{\infty}$. Furthermore, loophole-free violations remain possible, even when the efficiency at some sites is reduced well below 0.50.50.5, provided N>3N>3N>3.
A family of Bell-type inequalities is present, which are constructed directly from the "standard" Bell inequalities involving two dichotomic observables per site. It is shown that the inequalities are violated by all the generalized Greenberger-Horne-Zeilinger states of multiqubits. Remarkably, our new inequalities can provide stronger non-locality tests in a sense that the local reality inequalities are exponentially stronger than the corresponding multipartite separability inequalities. This reveals that the exponential violation of local realism by separable states is an interesting consequence of quantum fluctuation of multipartite systems.
Abstract In device-independent quantum information processing Bell inequalities are not only used as detectors of nonlocality, but also as certificates of relevant quantum properties. In order for these certificates to work, one very often needs Bell inequalities that are maximally violated by specific quantum states. Recently, in Salavrakos et al (2017 Phys. Rev. Lett. 119 040402) a general class of Bell inequalities, with arbitrary numbers of measurements and outcomes, has been designed, which are maximally violated by the maximally entangled states of two quantum systems of arbitrary dimension. In this work, we generalize these results to the multipartite scenario and obtain a general class of Bell inequalities maximally violated by the Greenberger–Horne–Zeilinger states of any number of parties and any local dimension. We then derive analytically their maximal quantum and nonsignaling values. We also obtain analytically the bound for detecting genuine nonlocality and compute the fully local bound for a few exemplary cases. Moreover, we consider the question of adapting this class of inequalities to partially entangled Greenberger–Horne–Zeilinger-like states for some special cases of low dimension and small number of parties. Through numerical methods, we find classes of inequalities maximally violated by these partially entangled states.
The nonclassical properties of the noisy three-qubit Greenberger-Horne-Zeilinger (GHZ) states, ${\ensuremath{\rho}}_{v}=v|\text{GHZ}\ensuremath{\rangle}\ensuremath{\langle}\text{GHZ}|+(1\ensuremath{-}v)\mathbb{1}/8$ parameterized by the visibility $0\ensuremath{\le}v\ensuremath{\le}1$ are investigated. Based on the violation of the $2\ifmmode\times\else\texttimes\fi{}2\ifmmode\times\else\texttimes\fi{}2$-setting Mermin inequality, ${\ensuremath{\rho}}_{v}$ is nonclassical for the parameter range $1/2<v\ensuremath{\le}1$. It has been posed whether additional settings would allow to lower the threshold visibility. Here we report on Bell inequalities giving a threshold value smaller than v=1/2v=1/2v=1/2. This rules out the possibility of a local hidden variable model in the limit of v=1/2v=1/2v=1/2. In particular, the lowest threshold visibility we found is v=0.496057v=0.496\phantom{\rule{0.16em}{0ex}}057v=0.496057, attainable with $5\ifmmode\times\else\texttimes\fi{}5\ifmmode\times\else\texttimes\fi{}5$ settings, whereas the most economical one in number of settings corresponds to $3\ifmmode\times\else\texttimes\fi{}3\ifmmode\times\else\texttimes\fi{}4$ settings. The method which enabled us to obtain these results, and in particular the about 10 000 tight Bell inequalities giving v<1/2v<1/2v<1/2, are also discussed in detail.
Abstract In device-independent quantum information processing Bell inequalities are not only used as detectors of nonlocality, but also as certificates of relevant quantum properties. In order for these certificates to work, one very often needs Bell inequalities that are maximally violated by specific quantum states. Recently, in Salavrakos et al (2017 Phys. Rev. Lett. 119 040402) a general class of Bell inequalities, with arbitrary numbers of measurements and outcomes, has been designed, which are maximally violated by the maximally entangled states of two quantum systems of arbitrary dimension. In this work, we generalize these results to the multipartite scenario and obtain a general class of Bell inequalities maximally violated by the Greenberger–Horne–Zeilinger states of any number of parties and any local dimension. We then derive analytically their maximal quantum and nonsignaling values. We also obtain analytically the bound for detecting genuine nonlocality and compute the fully local bound for a few exemplary cases. Moreover, we consider the question of adapting this class of inequalities to partially entangled Greenberger–Horne–Zeilinger-like states for some special cases of low dimension and small number of parties. Through numerical methods, we find classes of inequalities maximally violated by these partially entangled states.