  Quantum theory violates Bell's inequality, but not to the maximum extent that
is logically possible. We derive inequalities (generalizations of Cirel'son's
inequality) that quantify the upper bound of the violation, both for the
standard formalism and the formalism of generalized observables (POVMs). These
inequalities are quantum analogues of Bell inequalities, and they can be used
to test the quantum version of locality. We discuss the nature of this kind of
locality. We also go into the relation of our results to an argument by Popescu
and Rohrlich (Found. Phys. 24, 379 (1994)) that there is no general connection
between the existence of Cirel'son's bound and locality.
