  A very general local realistic theory of single $B^{0}_{d}$ mesons and of
correlated $B^{0}_{d}\bar{B^{0}_{d}}$ pairs is formulated. If these pairs are
produced in the $\Upsilon (4S)$ decay, the local realistic asymmetry for
observing pairs with like and unlike flavour at different proper times
remarkably differs from the quantum mechanical prediction. Asymmetric
B-factories provide a powerful tool for the study of the EPR problem since the
relative detection experiments are shown to be capable of a time-dependent
measurement precise enough to discriminate between local realism and quantum
theory.
