  It is well known that when a pair of random variables is statistically
independent, it has no-correlation (zero covariance, $E[XY] - E[X]E[Y] = 0$),
and that the converse is not true. However, if both of these random variables
take only two values, no-correlation entails statistical independence. We
provide here a general proof. We subsequently examine whether this equivalence
property carries over to quantum mechanical systems. A counter-example is
explicitly constructed to show that it does not. This observation provides yet
another simple theorem separating classical and quantum theories.
