Maximal violation of Bell's inequalities and Pauli spin matrices

The paper deals with the structure of Bell's inequalities (Clauser, Horne, Shimony, and Holt (CHSH) form). It is established that root 2 is the universal bound for Bell's correlations given by a general correlation duality on a complex linear space. Using geometric arguments the maximal violation of Bell's inequalities is described on this abstract level. It is proved that Bell's inequalities are maximally violated for general *-algebras and a faithful state exactly when the corresponding elements are the Pauli spin matrices. Interesting structural consequences of this result are derived. They demonstrate that the maximal violation requires a very "nonclassical" position of the local systems. Moreover, the maximal strength of Bell's correlations for a general state with respect to nearly commuting subalgebras is characterized in terms of the spin systems and tracial properties of the local states. (C) 2009 American Institute of Physics. [DOI: 10.1063/1.3190118]