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

AIP conference proceedings

The Reduction of Density Matrix and Measurement of Bell-CHSH Inequalities

V. A. Andreev

Published Jan 1, 2007• 0 citations
Publisher

Abstract

The right hand side of Bell inequalities don’t depend on a form of any two‐particle spin state. In the case of the generalized Bell‐Clauser‐Horne‐Schimony‐Holt inequalities the right hand side depend on a form of concrete two‐particle state. The left hand sides of these inequalities depend on four arbitrary vectors defined in three‐dimensional space. They define directions on which spins of particles forming a correlated pair are projected. Our aim is to find such vectors that left‐hand side of the inequality should takes its maximum value. In other words on these vectors the inequality transforms into equality. It is shown that it can be done with the help of special reduction of density matrix of two‐state spin state.

DOI
10.1063/1.2713465

Related papers

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

Reduction of the two-spin state density matrix and evaluation of the left-hand side of the Bell-CHSH inequality

V. A. Andreev

Journal of Russian Laser Research•Jul 1, 2006•7 citations

Spin quantum correlations of relativistic particles

Pablo L. Saldanha, Vlatko Vedral

We show that a pair of massive relativistic spin-1/2 particles prepared in a maximally entangled spin state in general is not capable of maximally violating the Clauser-Horne-Shimony-Holt (CHSH) version of Bell's inequalities without a postselection of the particles' momenta, representing a major difference in relation to nonrelativistic systems. This occurs because the quantization axis of the measurements performed on each particle depends on the particle velocity, such that it is not possible to define a reduced density matrix for the particles' spin. We also show that the amount of violation of the CHSH inequality depends on the reference frame and that in some frames the inequality may not be violated.

arXiv:1112.1272•Physical Review A•Jun 1, 2012•34 citations

Quantum Tomography and Verification of Generalized Bell-CHSH Inequalities

V. A. Andreev

It is shown that verification procedure of Bell inequalities consists of three parts. The first stage of verification procedure is consist of measurements of a set of spin correlations. Using them one can reconstruct a density matrix of two‐particle state. It can be done with the help of quantum tomography technique. At the second stage one must construct a generalized Bell inequality corresponding to this state. This inequality must be constructed in two forms: within the framework of the traditional formulation of quantum mechanics and in the frames of a local hidden variables theory. Only after that the results of experimental measurements can be compared with theoretical predictions. It is a third stage of verification.

AIP conference proceedings•Jan 1, 2005•1 citation

Remarks on the Bell-Clauser-Horne-Shimony-Holt inequality

Silvio Paolo sorella

We discuss the relationship between the Bogoliubov transformations, squeezed states, entanglement, and maximum violation of the Bell-Clauser-Horne-Shimony-Holt (Bell-CHSH) inequality. In particular, we point out that the construction of the four bounded operators entering the Bell-CHSH inequality can be worked out in a simple and general way, covering a large variety of models, ranging from quantum mechanism to relativistic quantum field theories. Various examples are employed to illustrate the above mentioned framework. We start by considering a pair of entangled spin-one particles and a squeezed oscillator in quantum mechanics, moving then to the relativistic complex quantum scalar field and to the analysis of the vacuum state in Minkowski space-time in terms of the left and right Rindler modes. In the latter case, the Bell-CHSH inequality turns out to be parametrized by the Unruh temperature.

arXiv:2301.02245•Physical review. D/Physical review. D.•Jun 14, 2023•5 citations

Bell's-type inequalities revisited: new constraints from objective reality

Héctor A. Múnera

The empirical evidence tends to confirm the quantum - mechanically (QM) predicted violation of Bell's - type inequalities. The latter are widely accepted to be correct representation of the objective reality (OR) underlying EPR's gedankenexperiment. Contrariwise, we argue here that the starting premises leading to Bell's and CHSH's inequalities are not appropriate formulations of OR in the EPR context. Indeed, the essence of EPR argument is the correlation between pair of particles travelling with some well defined orientations. To measure such correlation, one requires detectors with high resolution, and some appropriately defined relative orientations. On the contrary, experimental tests of Bell's inequalities typically involve counting the number of particles with some value of spin (or polarisation) using low-resolution detectors and completely arbitrary orientations. Depending upon the relative orientation of good-resolution detectors, we obtain two families of CHSH-type inequalities. One of these families is the most often used version of CHSH inequality, that was derived by Clauser and Horne with the supplementary no - enhancement assumption: here, it is obtained without special provisions. The other family is completely new and refers to coincidence counts from multiple events. Experimental test to distinguish between the QM and OR descriptions of nature should take into account these new constraints.

MOMENTO•Jul 1, 2002•0 citations
The right hand side of Bell inequalities don’t depend on a form of any two‐particle spin state. In the case of the generalized Bell‐Clauser‐Horne‐Schimony‐Holt inequalities the right hand side depend on a form of concrete two‐particle state. The left hand sides of these inequalities depend on four arbitrary vectors defined in three‐dimensional space. They define directions on which spins of particles forming a correlated pair are projected. Our aim is to find such vectors that left‐hand side of the inequality should takes its maximum value. In other words on these vectors the inequality transforms into equality. It is shown that it can be done with the help of special reduction of density matrix of two‐state spin state.
We show that a pair of massive relativistic spin-1/2 particles prepared in a maximally entangled spin state in general is not capable of maximally violating the Clauser-Horne-Shimony-Holt (CHSH) version of Bell's inequalities without a postselection of the particles' momenta, representing a major difference in relation to nonrelativistic systems. This occurs because the quantization axis of the measurements performed on each particle depends on the particle velocity, such that it is not possible to define a reduced density matrix for the particles' spin. We also show that the amount of violation of the CHSH inequality depends on the reference frame and that in some frames the inequality may not be violated.
It is shown that verification procedure of Bell inequalities consists of three parts. The first stage of verification procedure is consist of measurements of a set of spin correlations. Using them one can reconstruct a density matrix of two‐particle state. It can be done with the help of quantum tomography technique. At the second stage one must construct a generalized Bell inequality corresponding to this state. This inequality must be constructed in two forms: within the framework of the traditional formulation of quantum mechanics and in the frames of a local hidden variables theory. Only after that the results of experimental measurements can be compared with theoretical predictions. It is a third stage of verification.
We discuss the relationship between the Bogoliubov transformations, squeezed states, entanglement, and maximum violation of the Bell-Clauser-Horne-Shimony-Holt (Bell-CHSH) inequality. In particular, we point out that the construction of the four bounded operators entering the Bell-CHSH inequality can be worked out in a simple and general way, covering a large variety of models, ranging from quantum mechanism to relativistic quantum field theories. Various examples are employed to illustrate the above mentioned framework. We start by considering a pair of entangled spin-one particles and a squeezed oscillator in quantum mechanics, moving then to the relativistic complex quantum scalar field and to the analysis of the vacuum state in Minkowski space-time in terms of the left and right Rindler modes. In the latter case, the Bell-CHSH inequality turns out to be parametrized by the Unruh temperature.
The empirical evidence tends to confirm the quantum - mechanically (QM) predicted violation of Bell's - type inequalities. The latter are widely accepted to be correct representation of the objective reality (OR) underlying EPR's gedankenexperiment. Contrariwise, we argue here that the starting premises leading to Bell's and CHSH's inequalities are not appropriate formulations of OR in the EPR context. Indeed, the essence of EPR argument is the correlation between pair of particles travelling with some well defined orientations. To measure such correlation, one requires detectors with high resolution, and some appropriately defined relative orientations. On the contrary, experimental tests of Bell's inequalities typically involve counting the number of particles with some value of spin (or polarisation) using low-resolution detectors and completely arbitrary orientations. Depending upon the relative orientation of good-resolution detectors, we obtain two families of CHSH-type inequalities. One of these families is the most often used version of CHSH inequality, that was derived by Clauser and Horne with the supplementary no - enhancement assumption: here, it is obtained without special provisions. The other family is completely new and refers to coincidence counts from multiple events. Experimental test to distinguish between the QM and OR descriptions of nature should take into account these new constraints.