Dynamic violation of Bell's inequalities in the angular momentum representation

TL;DR

This paper introduces a new angular momentum-based parametrization of density matrices to analyze Bell inequality violations in composite quantum systems, demonstrating maximal violations and their relation to entanglement.

Contribution

It develops a novel angular momentum operator framework for density matrices and explores Bell inequality violations in qubit-qubit and qubit-qutrit systems, including time-dependent scenarios.

Findings

Violations of Bell inequalities can reach the Cirel'son limit.

A correlation between entanglement and Bell violation strength is established.

Periodic maximal Bell violations are observed in time-dependent states.

Abstract

A parametrization of density matrices of $d$ dimensions in terms of the raising $J_+$ and lowering $J_-$ angular momentum operators is established together with an implicit connection with the generalized Bloch-GellMann parameters. A general expression for the density matrix of the composite system of angular momenta $j_1$ and $j_2$ is obtained. In this matrix representation violations of the Bell-Clauser-Horne-Shimony-Holt inequalities are established for the $X$-states of a qubit-qubit, pure and mixed, composite system, as well as for a qubit-qutrit density matrix. In both cases maximal violation of the Bell inequalities can be reached, i.e., the Cirel'son limit. A correlation between the entanglement measure and a strong violation of the Bell factor is also given. For the qubit-qutrit composite system a time-dependent convex combination of the density matrix of the eigenstates of a two-particle Hamiltonian system is used to determine periodic maximal violations of the Bell's inequality.