Geometric chained inequalities for higher-dimensional systems

For systems of an arbitrary dimension, a theory of geometric chained Bell inequalities is presented. The approach is based on chained inequalities derived by Pykacz and Santos. For maximally entangled states, the inequalities lead to a complete $0=1$ contradiction with quantum predictions. Local realism suggests that the probability for the two observers to have identical results is 1 (that is, a perfect correlation is predicted), whereas quantum formalism gives an opposite prediction: the local results always differ. This is so for any dimension. We also show that with the inequalities, one can have a version of Bell's theorem which involves only correlations arbitrarily close to perfect ones.