Reduction of the density matrix and generalized bell inequalities

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