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.
Highlights
Differences and boundaries between the classical and quantum mechanics, as manifested by the Bell’s inequalities[1, 2], have been under active research investigations both theoretically and experimentally. We present yet another simple theorem which separates classical and quantum probability theories
We can have a pair of random variables which is not correlated but not statistically independent, and such examples can be constructed as well. (A known exception is when their joint probability density function has the form of a bivariate normal distribution[4].)
Our main statement here is that the converse is true, i.e., (4) and (5) are equivalent when both of these random variable take two finite distinct values
Summary
Differences and boundaries between the classical and quantum mechanics, as manifested by the Bell’s inequalities[1, 2], have been under active research investigations both theoretically and experimentally. We present yet another simple theorem which separates classical and quantum probability theories. It is well known and can be proven that when two random variables are statistically independent, they are not correlated. We can have a pair of random variables which is not correlated but not statistically independent, and such examples can be constructed as well. First show that when both of these random variables are not continuous and take only two distinct values, statistical independence and nocorrelation become equivalent in general. The above-mentioned theorem about two-state systems does not carry over to quantum mechanics We show this by explicitly constructing a counter-example
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