On Statistical Independence and No-Correlation for a Pair of Random Variables Taking Two Values: Classical and Quantum

TL;DR

This paper proves that for two-valued classical variables, no-correlation implies independence, but this equivalence does not hold in quantum systems, highlighting a fundamental difference between classical and quantum correlations.

Contribution

It establishes a general proof that no-correlation implies independence for two-valued classical variables and demonstrates the failure of this equivalence in quantum systems with a counter-example.

Findings

No-correlation implies independence for two-valued classical variables.

Counter-example shows the equivalence fails in quantum systems.

Highlights a fundamental difference between classical and quantum correlations.

Abstract

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.