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:1402.0378 · Journal of Physics A Mathematical and Theoretical

Optimization of Bell inequalities with invariant Tsirelson bound

Michael Epping, Hermann Kampermann, Dagmar Bruß

Published Oct 9, 2014• 3 citations
arXivHTMLPDF

Abstract

We consider a subclass of bipartite CHSH-type Bell inequalities. We investigate operations which leave their Tsirelson bound invariant, but change their classical bound. The optimal observables are unaffected except for a relative rotation of the two laboratories. We illustrate the utility of these operations by giving explicit examples. We prove that, for a fixed quantum state and fixed measurement setup except for a relative rotation of the two laboratories, there is a Bell inequality that is maximally violated for this rotation, and we optimize some Bell inequalities with respect to the maximal violation. Finally, we optimize the qutrit to qubit ratio of some dimension witnessing Bell inequalities. This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to '50 years of Bell's theorem'.

DOI
10.1088/1751-8113/47/42/424015

Related papers

Tracing the bounds on Bell-type inequalities

Stefan Filipp, Karl Svozil

Bell-type inequalities and violations thereof reveal the fundamental differences between standard probability theory and its quantum counterpart. In the course of previous investigations ultimate bounds on quantum mechanical violations have been found. For example, Tsirelson's bound constitutes a global upper limit for quantum violations of the Clauser-Horne-Shimony-Holt (CHSH) and the Clauser-Horne (CH) inequalities. Here we investigate a method for calculating the precise quantum bounds on arbitrary Bell-type inequalities by solving the eigenvalue problem for the operator associated with each Bell-type inequality. Thereby, we use the min-max principle to calculate the norm of these self-adjoint operators from the maximal eigenvalue yielding the upper bound for a particular set of measurement parameters. The eigenvectors corresponding to the maximal eigenvalues provide the quantum state for which a Bell-type inequality is maximally violated.

•Jan 1, 2004•9 citations

Tracing the bounds on Bell-type inequalities

Stefan Filipp

Bell‐type inequalities and violations thereof reveal the fundamental differences between standard probability theory and its quantum counterpart. In the course of previous investigations ultimate bounds on quantum mechanical violations have been found. For example, Tsirelson’s bound constitutes a global upper limit for quantum violations of the Clauser‐Horne‐Shimony‐Holt (CHSH) and the Clauser‐Horne (CH) inequalities. Here we investigate a method for calculating the precise quantum bounds on arbitrary Bell‐type inequalities by solving the eigenvalue problem for the operator associated with each Bell‐type inequality. Thereby, we use the min‐max principle to calculate the norm of these self‐adjoint operators from the maximal eigenvalue yielding the upper bound for a particular set of measurement parameters. The eigenvectors corresponding to the maximal eigenvalues provide the quantum state for which a Bell‐type inequality is maximally violated.

arXiv:quant-ph/0407145•AIP conference proceedings•Jan 1, 2005•5 citations

From Bell Inequalities to Tsirelson's Theorem

David Avis, Sonoko Moriyama, Masaki Owari

The first part of this paper contains an introduction to Bell inequalities and Tsirelson's theorem for the non-specialist. The next part gives an explicit optimum construction for the “hard” part of Tsirelson's theorem. In the final part we describe how upper bounds on the maximal quantum violation of Bell inequalities can be obtained by an extension of Tsirelson's theorem, and survey very recent results on how exact bounds may be obtained by solving an infinite series of semidefinite programs.

arXiv:0812.4887•IEICE Transactions on Fundamentals of Electronics Communications and Computer Sciences•Jan 1, 2009•12 citations

Translationally invariant multipartite Bell inequalities involving only two-body correlators

J Tura, A B Sainz, T Vértesi +3 more

Bell inequalities are natural tools that allow one to certify the presence of nonlocality in quantum systems. The known constructions of multipartite Bell inequalities contain, however, correlation functions involving all observers, making their experimental implementation difficult. The main purpose of this work is to explore the possibility of witnessing nonlocality in multipartite quantum states from the easiest-to-measure quantities, that is, the two-body correlations. In particular, we determine all three- and four-partite Bell inequalities constructed from one- and two-body expectation values that obey translational symmetry, and show that they reveal nonlocality in multipartite states. Also, by providing a particular example of a five-partite Bell inequality, we show that nonlocality can be detected from two-body correlators involving only nearest neighbours. Finally, we demonstrate that any translationally invariant Bell inequality can be maximally violated by a translationally invariant state and the same set of observables at all sites. We provide a numerical algorithm allowing one to seek for maximal violation of a translationally invariant Bell inequality. This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to ‘50 years of Bell’s theorem’.

arXiv:1312.0265•Journal of Physics A Mathematical and Theoretical•Oct 9, 2014•32 citations

Generalizing Tsirelson's Bound on Bell Inequalities Using a Min-Max Principle

Stefan Filipp, Karl Svozil

Bounds on the norm of quantum operators associated with classical Bell-type inequalities can be derived from their maximal eigenvalues. This quantitative method enables detailed predictions of the maximal violations of Bell-type inequalities.

arXiv:quant-ph/0403175•Physical Review Letters•Sep 23, 2004•37 citations

Relations between entanglement witnesses and Bell inequalities

Philipp Hyllus, Otfried Gühne, Dagmar Bruß +1 more

Bell inequalities, considered within quantum mechanics, can be regarded as nonoptimal witness operators. We discuss the relationship between such Bell witnesses and general entanglement witnesses in detail for the Bell inequality derived by Clauser, Horne, Shimony, and Holt (CHSH) [Phys. Rev. Lett. 23, 880 (1969)]. We derive bounds on how much an optimal witness has to be shifted by adding the identity operator to make it positive on all states admitting a local hidden variable model. In the opposite direction, we obtain tight bounds for the maximal proportion of the identity operator that can be subtracted from such a CHSH witness, while preserving the witness properties. Finally, we investigate the structure of CHSH witnesses directly by relating their diagonalized form to optimal witnesses of two different classes.

arXiv:quant-ph/0504079•Physical Review A•Jul 20, 2005•112 citations
We consider a subclass of bipartite CHSH-type Bell inequalities. We investigate operations which leave their Tsirelson bound invariant, but change their classical bound. The optimal observables are unaffected except for a relative rotation of the two laboratories. We illustrate the utility of these operations by giving explicit examples. We prove that, for a fixed quantum state and fixed measurement setup except for a relative rotation of the two laboratories, there is a Bell inequality that is maximally violated for this rotation, and we optimize some Bell inequalities with respect to the maximal violation. Finally, we optimize the qutrit to qubit ratio of some dimension witnessing Bell inequalities. This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to '50 years of Bell's theorem'.
Bell-type inequalities and violations thereof reveal the fundamental differences between standard probability theory and its quantum counterpart. In the course of previous investigations ultimate bounds on quantum mechanical violations have been found. For example, Tsirelson's bound constitutes a global upper limit for quantum violations of the Clauser-Horne-Shimony-Holt (CHSH) and the Clauser-Horne (CH) inequalities. Here we investigate a method for calculating the precise quantum bounds on arbitrary Bell-type inequalities by solving the eigenvalue problem for the operator associated with each Bell-type inequality. Thereby, we use the min-max principle to calculate the norm of these self-adjoint operators from the maximal eigenvalue yielding the upper bound for a particular set of measurement parameters. The eigenvectors corresponding to the maximal eigenvalues provide the quantum state for which a Bell-type inequality is maximally violated.
Bell‐type inequalities and violations thereof reveal the fundamental differences between standard probability theory and its quantum counterpart. In the course of previous investigations ultimate bounds on quantum mechanical violations have been found. For example, Tsirelson’s bound constitutes a global upper limit for quantum violations of the Clauser‐Horne‐Shimony‐Holt (CHSH) and the Clauser‐Horne (CH) inequalities. Here we investigate a method for calculating the precise quantum bounds on arbitrary Bell‐type inequalities by solving the eigenvalue problem for the operator associated with each Bell‐type inequality. Thereby, we use the min‐max principle to calculate the norm of these self‐adjoint operators from the maximal eigenvalue yielding the upper bound for a particular set of measurement parameters. The eigenvectors corresponding to the maximal eigenvalues provide the quantum state for which a Bell‐type inequality is maximally violated.
The first part of this paper contains an introduction to Bell inequalities and Tsirelson's theorem for the non-specialist. The next part gives an explicit optimum construction for the “hard” part of Tsirelson's theorem. In the final part we describe how upper bounds on the maximal quantum violation of Bell inequalities can be obtained by an extension of Tsirelson's theorem, and survey very recent results on how exact bounds may be obtained by solving an infinite series of semidefinite programs.
Bell inequalities are natural tools that allow one to certify the presence of nonlocality in quantum systems. The known constructions of multipartite Bell inequalities contain, however, correlation functions involving all observers, making their experimental implementation difficult. The main purpose of this work is to explore the possibility of witnessing nonlocality in multipartite quantum states from the easiest-to-measure quantities, that is, the two-body correlations. In particular, we determine all three- and four-partite Bell inequalities constructed from one- and two-body expectation values that obey translational symmetry, and show that they reveal nonlocality in multipartite states. Also, by providing a particular example of a five-partite Bell inequality, we show that nonlocality can be detected from two-body correlators involving only nearest neighbours. Finally, we demonstrate that any translationally invariant Bell inequality can be maximally violated by a translationally invariant state and the same set of observables at all sites. We provide a numerical algorithm allowing one to seek for maximal violation of a translationally invariant Bell inequality. This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to ‘50 years of Bell’s theorem’.
Bell inequalities, considered within quantum mechanics, can be regarded as nonoptimal witness operators. We discuss the relationship between such Bell witnesses and general entanglement witnesses in detail for the Bell inequality derived by Clauser, Horne, Shimony, and Holt (CHSH) [Phys. Rev. Lett. 23, 880 (1969)]. We derive bounds on how much an optimal witness has to be shifted by adding the identity operator to make it positive on all states admitting a local hidden variable model. In the opposite direction, we obtain tight bounds for the maximal proportion of the identity operator that can be subtracted from such a CHSH witness, while preserving the witness properties. Finally, we investigate the structure of CHSH witnesses directly by relating their diagonalized form to optimal witnesses of two different classes.