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arXiv · 1804.09248

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

Toru Ohira
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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.

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Toru Ohira. 2018-04-21. On Statistical Independence and No-Correlation for a Pair of Random Variables Taking Two Values: Classical and Quantum. https://doi.org/10.1093/ptep/pty085

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