© 2026 Aqora Quantum S.A.S.

Terms Privacy Legal Notice

arXiv:1112.2827 · Physical Review A

Homogenization of Bell inequalities

Yu-Chun Wu, Marek Żukowski

Published Feb 21, 2012• 3 citations

Abstract

A technique, which we call homogenization, is applied to transform Clauser and Horne- (CH)-type Bell inequalities that contain lower-order correlations into CH, Shimony, and Holt- (CHSH)-type Bell inequalities, which are defined for highest-order correlation functions. A homogenization leads to inequalities involving more settings, that is, a choice of one more observable is possible for each party. We show that this technique preserves the tightness of Bell inequalities: A homogenization of a tight CH-type Bell inequality is still a tight CHSH-type Bell inequality. As an example, we obtain $3\ifmmode\times\else\texttimes\fi{}3\ifmmode\times\else\texttimes\fi{}3$ CHSH-type Bell inequalities by the homogenization of $2\ifmmode\times\else\texttimes\fi{}2\ifmmode\times\else\texttimes\fi{}2$ CH-type Bell inequalities derived by Sliwa [Phys. Lett. A 317, 165 (2003)].

DOI: 10.1103/physreva.85.022119

Related papers

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.

arXiv:1302.6698•Physical Review A•Aug 29, 2013•11 citations

Clauser-Horne-Shimony-Holt-type Bell inequalities involving a party with two or three local binary settings

Yu-Chun Wu, Piotr Badziąg, Marek Żukowski

We construct a simple algorithm to generate any Clauser-Horne-Shimony-Holt- (CHSH-) type Bell inequality involving a party with two local binary measurements from two CHSH-type inequalities without this party. The algorithm readily generalizes to situations where the additional observer uses three measurement settings. There, each inequality involving the additional party is constructed from three inequalities with this party excluded. With this generalization at hand, we construct and analyze a class of symmetric inequalities for four observers and three experimental settings per observer.

arXiv:0810.1169•Physical Review A•Feb 11, 2009•8 citations

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

Tight correlation-function Bell inequality for multipartite d -dimensional systems

Jing‐Ling Chen, Dong-Ling Deng

We generalize the correlation functions of the Clauser-Horne-Shimony-Holt (CHSH) inequality to multipartite $d$-dimensional systems. All the Bell inequalities based on this generalization take the same simple form as the CHSH inequality. For small systems, numerical results show that the new inequalities are tight and we believe this is also valid for higher-dimensional systems. Moreover, the new inequalities are relevant to the previous ones and, for bipartite systems, our inequality is equivalent to the Collins-Gisin-Linden-Masser-Popescu inequality.

arXiv:0809.1713•Physical Review A•Jan 23, 2009•11 citations

A logical loophole in the derivation of the CHSH Bell-type inequality

G. Coddens

We point out a loophole in the derivation of the Bell inequalities in the form proposed by Clauser, Horne, Shimony and Holt (CHSH). In this derivation it is assumed that statistical independence is a necessary consequence of locality, but this is not a logical necessity.

arXiv:1309.6057•Sep 24, 2013•0 citations

A Bell inequality which can be used to test locality more simply than Clauser-Horne inequality and which is violated by a larger magnitude of violation than Clauser-Horne-Shimony-Holt inequality

M. Ardehali

A correlation inequality is derived from local realism and a supplementary assumption. Unlike Clauser-Horne (CH) inequality [or Clauser-Horne-Shimony-Holt (CHSH) inequality] which is violated by quantum mechanics by a factor of $\sqrt 2$, this inequality is violated by a factor of 1.5. Thus the magnitude of violation of this inequality is approximately 20.7% larger than the magnitude of violation of previous inequalities. Moreover, unlike CH (or CHSH) inequality which requires the measurement of five detection probabilities, the present inequality requires the measurement of only two detection probabilities. This inequality can therefore be used to test locality more simply than CH or CHSH inequality.

arXiv:quant-ph/9709043•Sep 19, 1997•0 citations

A technique, which we call homogenization, is applied to transform Clauser and Horne- (CH)-type Bell inequalities that contain lower-order correlations into CH, Shimony, and Holt- (CHSH)-type Bell inequalities, which are defined for highest-order correlation functions. A homogenization leads to inequalities involving more settings, that is, a choice of one more observable is possible for each party. We show that this technique preserves the tightness of Bell inequalities: A homogenization of a tight CH-type Bell inequality is still a tight CHSH-type Bell inequality. As an example, we obtain

3\ifmmode\times\else\texttimes\fi{}3\ifmmode\times\else\texttimes\fi{}3

CHSH-type Bell inequalities by the homogenization of

2\ifmmode\times\else\texttimes\fi{}2\ifmmode\times\else\texttimes\fi{}2

CH-type Bell inequalities derived by Sliwa [Phys. Lett. A 317, 165 (2003)].

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.

We construct a simple algorithm to generate any Clauser-Horne-Shimony-Holt- (CHSH-) type Bell inequality involving a party with two local binary measurements from two CHSH-type inequalities without this party. The algorithm readily generalizes to situations where the additional observer uses three measurement settings. There, each inequality involving the additional party is constructed from three inequalities with this party excluded. With this generalization at hand, we construct and analyze a class of symmetric inequalities for four observers and three experimental settings per observer.

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.

We generalize the correlation functions of the Clauser-Horne-Shimony-Holt (CHSH) inequality to multipartite

d

-dimensional systems. All the Bell inequalities based on this generalization take the same simple form as the CHSH inequality. For small systems, numerical results show that the new inequalities are tight and we believe this is also valid for higher-dimensional systems. Moreover, the new inequalities are relevant to the previous ones and, for bipartite systems, our inequality is equivalent to the Collins-Gisin-Linden-Masser-Popescu inequality.

A correlation inequality is derived from local realism and a supplementary assumption. Unlike Clauser-Horne (CH) inequality [or Clauser-Horne-Shimony-Holt (CHSH) inequality] which is violated by quantum mechanics by a factor of

\sqrt 2

, this inequality is violated by a factor of 1.5. Thus the magnitude of violation of this inequality is approximately 20.7% larger than the magnitude of violation of previous inequalities. Moreover, unlike CH (or CHSH) inequality which requires the measurement of five detection probabilities, the present inequality requires the measurement of only two detection probabilities. This inequality can therefore be used to test locality more simply than CH or CHSH inequality.