On statistical independence and no-correlation for a pair of random variables taking two values: Classical and quantum

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.