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:1301.4823

A note on the geometric interpretation of Bell's inequalities

Paolo Dai Pra, Michele Pavon, Neeraja Sahasrabudhe

Published Jan 21, 2013• 0 citations
arXivHTMLPDF

Abstract

Using results of Pitowsky and Gupta, we show in a direct, elementary fashion that, in the case of three spins, Bell's inequalities indeed provide a representation of the tetrahedron of all spin correlation matrices as intersection of half-spaces.
DOI
10.48550/arxiv.1301.4823

Related papers

A Note on the Geometric Interpretation of Bell’s Inequalities

Paolo Dai Pra, Michele Pavon, Neeraja Sahasrabudhe

Letters in Mathematical Physics•May 30, 2013•4 citations

From breakfast, lunch, and dinner to the Bell-inequality tetrahedron

Francesco Giacosa

A simple pedagogical discussion and visualization of certain Bell inequalities is presented with the help of a system involving two \textquotedblleft twins\textquotedblright, originally discussed in 2605.03104. These twins answer identically when the same `question' is posed, here exemplified with: Did you like breakfast/lunch/dinner? In analogy, for two entangled particles at distant locations, the `twin-like' quantum state is such that the same outcome for the same measurement (i.e. spin direction) is obtained. In the space $(X,Y,Z)$ of mixed moments (i.e. $X$ refers to breakfast-lunch correlation, being $1$ for same answers and $-1$ for opposite ones), the space of Bell local models is contained in a tetrahedron, while the quantum space is bound by an elliptope that embeds the tetrahedron. The whole no-signalling region in the $(X,Y,Z)$ space is the cube $[-1,1]^{3}.$ The twin scenario applies e.g. to the decay of the Higgs into fermions.

quant-pharXiv:2609.14436•Sep 15, 2026•0 citations

Two-particle spin states and generalized Bell’s inequalities

V. A. Andreev, Vladimir I. Man’ko

A new formalism for calculation of the spin correlation magnitude is developed. In this approach, the mean value of an operator acting in the four-dimensional space of two-particle spin states has the form of a scalar product of vectors defined in the three-dimensional direction space. A complete description of the two-particle spin states satisfying Bell inequalities is given in this formalism. It is shown that these states include both factorable and entangled states.

Journal of Experimental and Theoretical Physics Letters•Jul 1, 2000•14 citations

Breaking of Bell inequality based on $S_4$ symmetry

Katarzyna Bolonek-Lasoń

In two recent papers (Phys. Rev. A90 (2014), 062121 and Phys. Rev. A91 (2015), 052110) an interesting method of analyzing the breaking of Bell inequalities has been proposed which is based on the theory of finite group representations. We apply here this method to more complicated example of $S_4$ symmetry. We show how the Bell inequality can be related to the symmetries of regular tetrahedron. By choosing the orbits of threedimensional representation of $S_4$ determined by the geometry of tetrahedron we find that the Bell inequality under consideration is broken in quantum theory. The corresponding nonlocal game is analyzed.

•Mar 24, 2016•1 citation

Bell’s theorem without inequalities

Daniel M. Greenberger, Michael Horne, Abner Shimony +1 more

It is demonstrated that the premisses of the Einstein–Podolsky–Rosen paper are inconsistent when applied to quantum systems consisting of at least three particles. The demonstration reveals that the EPR program contradicts quantum mechanics even for the cases of perfect correlations. By perfect correlations is meant arrangements by which the result of the measurement on one particle can be predicted with certainty given the outcomes of measurements on the other particles of the system. This incompatibility with quantum mechanics is stronger than the one previously revealed for two-particle systems by Bell’s inequality, where no contradiction arises at the level of perfect correlations. Both spin-correlation and multiparticle interferometry examples are given of suitable three- and four-particle arrangements, both at the gedanken and at the real experiment level.

American Journal of Physics•Dec 1, 1990•2,402 citations

Violation of Bell inequality based on S4 symmetry

Katarzyna Bolonek-Lasoń

In two recent papers [Phys. Rev. A 90, 062121 (2014) and Phys. Rev. A, 91, 052110 (2015)] an interesting method of analyzing the violation of Bell inequalities has been proposed which is based on the theory of finite group representations. Here we apply this method to more complicated examples of ${S}_{4}$ symmetry. We show how the Bell inequality can be related to the symmetries of regular tetrahedron. By choosing the orbits of three-dimensional representations of ${S}_{4}$ determined by the geometry of tetrahedron we find that the Bell inequality under consideration is violated in quantum theory. The corresponding nonlocal game is analyzed.

arXiv:1603.07740•Physical Review A•Aug 9, 2016•6 citations
A simple pedagogical discussion and visualization of certain Bell inequalities is presented with the help of a system involving two \textquotedblleft twins\textquotedblright, originally discussed in 2605.03104. These twins answer identically when the same `question' is posed, here exemplified with: Did you like breakfast/lunch/dinner? In analogy, for two entangled particles at distant locations, the `twin-like' quantum state is such that the same outcome for the same measurement (i.e. spin direction) is obtained. In the space (X,Y,Z)(X,Y,Z)(X,Y,Z) of mixed moments (i.e. XXX refers to breakfast-lunch correlation, being 111 for same answers and −1-1−1 for opposite ones), the space of Bell local models is contained in a tetrahedron, while the quantum space is bound by an elliptope that embeds the tetrahedron. The whole no-signalling region in the (X,Y,Z)(X,Y,Z)(X,Y,Z) space is the cube [−1,1]3.[-1,1]^{3}.[−1,1]3. The twin scenario applies e.g. to the decay of the Higgs into fermions.
Breaking of Bell inequality based on S4S_4S4​ symmetry
In two recent papers (Phys. Rev. A90 (2014), 062121 and Phys. Rev. A91 (2015), 052110) an interesting method of analyzing the breaking of Bell inequalities has been proposed which is based on the theory of finite group representations. We apply here this method to more complicated example of S4S_4S4​ symmetry. We show how the Bell inequality can be related to the symmetries of regular tetrahedron. By choosing the orbits of threedimensional representation of S4S_4S4​ determined by the geometry of tetrahedron we find that the Bell inequality under consideration is broken in quantum theory. The corresponding nonlocal game is analyzed.
It is demonstrated that the premisses of the Einstein–Podolsky–Rosen paper are inconsistent when applied to quantum systems consisting of at least three particles. The demonstration reveals that the EPR program contradicts quantum mechanics even for the cases of perfect correlations. By perfect correlations is meant arrangements by which the result of the measurement on one particle can be predicted with certainty given the outcomes of measurements on the other particles of the system. This incompatibility with quantum mechanics is stronger than the one previously revealed for two-particle systems by Bell’s inequality, where no contradiction arises at the level of perfect correlations. Both spin-correlation and multiparticle interferometry examples are given of suitable three- and four-particle arrangements, both at the gedanken and at the real experiment level.
In two recent papers [Phys. Rev. A 90, 062121 (2014) and Phys. Rev. A, 91, 052110 (2015)] an interesting method of analyzing the violation of Bell inequalities has been proposed which is based on the theory of finite group representations. Here we apply this method to more complicated examples of S4{S}_{4}S4​ symmetry. We show how the Bell inequality can be related to the symmetries of regular tetrahedron. By choosing the orbits of three-dimensional representations of S4{S}_{4}S4​ determined by the geometry of tetrahedron we find that the Bell inequality under consideration is violated in quantum theory. The corresponding nonlocal game is analyzed.