  Bell-type inequalities and violations thereof reveal the fundamental
differences between standard probability theory and its quantum counterpart. In
the course of previous investigations ultimate bounds on quantum mechanical
violations have been found. For example, Tsirelson's bound constitutes a global
upper limit for quantum violations of the Clauser-Horne-Shimony-Holt (CHSH) and
the Clauser-Horne (CH) inequalities. Here we investigate a method for
calculating the precise quantum bounds on arbitrary Bell-type inequalities by
solving the eigenvalue problem for the operator associated with each Bell-type
inequality. Thereby, we use the min-max principle to calculate the norm of
these self-adjoint operators from the maximal eigenvalue yielding the upper
bound for a particular set of measurement parameters. The eigenvectors
corresponding to the maximal eigenvalues provide the quantum state for which a
Bell-type inequality is maximally violated.
