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

arXiv:0802.0125 · Physical Review A

Gisin’s theorem for two d -dimensional systems based on the Collins-Gisin-Linden-Masser-Popescu inequality

Jing‐Ling Chen, Dong-Ling Deng, Ming-Guang Hu

Published Jun 17, 2008• 15 citations
arXivHTMLPDF

Abstract

In this Rapid Communication, we show analytically that all pure entangled states of two $d$-dimensional systems (qudits) violate the Collins-Gisin-Linden-Masser-Popescu (CGLMP) inequality. This property was pointed out by Gisin in the qubit case and then generalized by Gisin-Peres and Popescu-Rohrlich to the qudit case based on the Clauser-Horne-Shimony-Holt (CHSH) inequality. We report the Gisin's theorem for two qudits by making use of the CGLMP inequality.

DOI
10.1103/physreva.77.060306

Related papers

Multisetting Bell inequality for qudits

Se-Wan Ji, Jinhyoung Lee, James Lim +2 more

We propose a generalized Bell inequality for two three-dimensional systems with three settings in each local measurement. It is shown that this inequality is maximally violated if local measurements are configured to be mutually unbiased and a composite state is maximally entangled. This feature is similar to Clauser-Horne-Shimony-Holt inequality for two qubits but is in contrast with the two types of inequalities, Collins-Gisin-Linden-Massar-Popescu and Son-Lee-Kim, for high-dimensional systems. The generalization to aribitrary prime-dimensional systems is discussed.

arXiv:0810.2838•Physical Review A•Nov 6, 2008•44 citations

Gisin's theorem cannot be generalized to all multi-particle states

Marek Żukowski, Časlav Brukner, Wiesław Laskowski +1 more

Gisin's theorem states that any pure entangled state of two particles violates Clauser-Horne-Shimony-Holt inequality which is a bound on correlations between measurements on two particles obtained within local realistic picture. We find that this theorem cannot be generalized to all multi-particle states for the case where measurements on each particle can be chosen between two arbitrary dichotomic observables. In particular we find a family of pure entangled states of N qubits which do not violate any Bell inequality for correlations between such measurements. We also find that Mermin-Ardehali-Belinskii-Klyshko inequalities may not always be optimal for refutation of local realistic description of correlations.

•Oct 15, 2001•1 citation

Sufficient condition for nonexistence of symmetric extension of qudits using Bell inequalities

Meenu Kumari, Shohini Ghose, Robert B. Mann

We analyze the connection between Bell inequality violations and symmetric extendibility of quantum states. We prove that 2-qubit reduced states of multiqubit symmetric pure states do not violate the Bell Clauser-Horne-Shimony-Holt (CHSH) inequality. We then prove the more general converse that any 2-qubit state that violates the CHSH inequality cannot have a symmetric extension. We extend our analysis to qudits and provide a test for symmetric extendibility of 2-qudit states. We show that if a 2-qudit Bell inequality is monogamous, then any 2-qudit state that violates this inequality does not have a symmetric extension. For the specific case of 2-qutrit states, we use numerical evidence to conjecture that the Collins-Gisin-Linden-Massar-Popescu (CGLMP) inequality is monogamous. Hence, the violation of the CGLMP inequality by any 2-qutrit state could be a sufficient condition for the nonexistence of its symmetric extension.

arXiv:1704.06516•Physical Review A•Jul 31, 2017•3 citations

Maximal Violation of the Collins-Gisin-Linden-Massar-Popescu Inequality for Infinite Dimensional States

Stefan Zohren, Richard David Gill

We present a much simplified version of the Collins-Gisin-Linden-Massar-Popescu inequality for the $2\ifmmode\times\else\texttimes\fi{}2\ifmmode\times\else\texttimes\fi{}d$ Bell scenario. Numerical maximization of the violation of this inequality over all states and measurements suggests that the optimal state is far from maximally entangled, while the best measurements are the same as conjectured best measurements for the maximally entangled state. For very large values of $d$ the inequality seems to reach its minimal value given by the probability constraints. This gives numerical evidence for a tight quantum Bell inequality (or generalized Csirelson inequality) for the $2\ifmmode\times\else\texttimes\fi{}2\ifmmode\times\else\texttimes\fi{}\ensuremath{\infty}$ scenario.

arXiv:quant-ph/0612020•Physical Review Letters•Mar 27, 2008•113 citations

Survey on the Bell nonlocality of a pair of entangled qudits

Alejandro Fonseca, Anna de Rosier, Tamás Vértesi +2 more

The question of how Bell nonlocality behaves in bipartite systems of higher dimensions is addressed. By employing the probability of violation of local realism under random measurements as the figure of merit, we investigate the nonlocality of entangled qudits with dimensions ranging from $d=2$ up to $d=10$. We proceed in two complementary directions. First, we study the specific Bell scenario defined by the Collins-Gisin-Linden-Massar-Popescu (CGLMP) inequality. Second, we consider the nonlocality of the same states under a more general perspective by directly addressing the space of joint probabilities (computing the frequencies of behaviours outside the local polytope). In both approaches we find that the nonlocality decreases as the dimension $d$ grows, but in quite distinct ways. While the drop in the probability of violation is exponential in the CGLMP scenario, it presents, at most, a linear decay in the space of behaviors. Furthermore, in the latter approach the states that produce maximal numeric violations in the CGLMP inequality present low probabilities of violation in comparison to maximally entangled states, so no anomaly is observed. Finally, the nonlocality of states with nonmaximal Schmidt rank is investigated.

arXiv:1805.09451•Physical Review A•Oct 3, 2018•24 citations

Analytical proof of Gisin’s theorem for three qubits

Sujit K. Choudhary, Sibasish Ghosh A, Guruprasad Kar +1 more

Gisin's theorem assures that for any pure bipartite entangled state, there is violation of the inequality of Bell and of Clauser, Horne, Shimony, and Holt, revealing its contradiction with local realistic model. Whether a similar result holds for three-qubit pure entangled states remained unresolved. We show analytically that all three-qubit pure entangled states violate a Bell-type inequality, derived on the basis of local realism, by exploiting the Hardy's nonlocality argument.

arXiv:0901.1669•Physical Review A•Apr 14, 2010•26 citations
In this Rapid Communication, we show analytically that all pure entangled states of two ddd-dimensional systems (qudits) violate the Collins-Gisin-Linden-Masser-Popescu (CGLMP) inequality. This property was pointed out by Gisin in the qubit case and then generalized by Gisin-Peres and Popescu-Rohrlich to the qudit case based on the Clauser-Horne-Shimony-Holt (CHSH) inequality. We report the Gisin's theorem for two qudits by making use of the CGLMP inequality.
We propose a generalized Bell inequality for two three-dimensional systems with three settings in each local measurement. It is shown that this inequality is maximally violated if local measurements are configured to be mutually unbiased and a composite state is maximally entangled. This feature is similar to Clauser-Horne-Shimony-Holt inequality for two qubits but is in contrast with the two types of inequalities, Collins-Gisin-Linden-Massar-Popescu and Son-Lee-Kim, for high-dimensional systems. The generalization to aribitrary prime-dimensional systems is discussed.
Gisin's theorem states that any pure entangled state of two particles violates Clauser-Horne-Shimony-Holt inequality which is a bound on correlations between measurements on two particles obtained within local realistic picture. We find that this theorem cannot be generalized to all multi-particle states for the case where measurements on each particle can be chosen between two arbitrary dichotomic observables. In particular we find a family of pure entangled states of N qubits which do not violate any Bell inequality for correlations between such measurements. We also find that Mermin-Ardehali-Belinskii-Klyshko inequalities may not always be optimal for refutation of local realistic description of correlations.
We analyze the connection between Bell inequality violations and symmetric extendibility of quantum states. We prove that 2-qubit reduced states of multiqubit symmetric pure states do not violate the Bell Clauser-Horne-Shimony-Holt (CHSH) inequality. We then prove the more general converse that any 2-qubit state that violates the CHSH inequality cannot have a symmetric extension. We extend our analysis to qudits and provide a test for symmetric extendibility of 2-qudit states. We show that if a 2-qudit Bell inequality is monogamous, then any 2-qudit state that violates this inequality does not have a symmetric extension. For the specific case of 2-qutrit states, we use numerical evidence to conjecture that the Collins-Gisin-Linden-Massar-Popescu (CGLMP) inequality is monogamous. Hence, the violation of the CGLMP inequality by any 2-qutrit state could be a sufficient condition for the nonexistence of its symmetric extension.
We present a much simplified version of the Collins-Gisin-Linden-Massar-Popescu inequality for the $2\ifmmode\times\else\texttimes\fi{}2\ifmmode\times\else\texttimes\fi{}d$ Bell scenario. Numerical maximization of the violation of this inequality over all states and measurements suggests that the optimal state is far from maximally entangled, while the best measurements are the same as conjectured best measurements for the maximally entangled state. For very large values of ddd the inequality seems to reach its minimal value given by the probability constraints. This gives numerical evidence for a tight quantum Bell inequality (or generalized Csirelson inequality) for the $2\ifmmode\times\else\texttimes\fi{}2\ifmmode\times\else\texttimes\fi{}\ensuremath{\infty}$ scenario.
The question of how Bell nonlocality behaves in bipartite systems of higher dimensions is addressed. By employing the probability of violation of local realism under random measurements as the figure of merit, we investigate the nonlocality of entangled qudits with dimensions ranging from d=2d=2d=2 up to d=10d=10d=10. We proceed in two complementary directions. First, we study the specific Bell scenario defined by the Collins-Gisin-Linden-Massar-Popescu (CGLMP) inequality. Second, we consider the nonlocality of the same states under a more general perspective by directly addressing the space of joint probabilities (computing the frequencies of behaviours outside the local polytope). In both approaches we find that the nonlocality decreases as the dimension ddd grows, but in quite distinct ways. While the drop in the probability of violation is exponential in the CGLMP scenario, it presents, at most, a linear decay in the space of behaviors. Furthermore, in the latter approach the states that produce maximal numeric violations in the CGLMP inequality present low probabilities of violation in comparison to maximally entangled states, so no anomaly is observed. Finally, the nonlocality of states with nonmaximal Schmidt rank is investigated.