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

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.