Uncorrelatedness of $|X_1|^2$ and $|X_2|^2$, if $X_1\sim\mathcal{CN}(0,\sigma_X^2)$ and $X_2\sim\mathcal{CN}(0,\sigma_X^2)$ are uncorrelated

Question

Let $X_1$ and $X_1$ be uncorrelated identically distributed zero-mean complex Gaussian random variables, i.e., $X_1\sim\mathcal{CN}(0,\sigma_X^2)$ and $X_2\sim\mathcal{CN}(0,\sigma_X^2)$.

Knowing that $X_1$ and $X_2$ are uncorrelated, what can we say about exponentially distributed random variables $|X_1|^2$ and $|X_2|^2$? Are they uncorrelated, or we lost the uncorrelatedness?

What if $X_1$ and $X_2$ are independent instead? I believe the independence will be preserved between $|X_1|^2$ and $|X_2|^2$.

Answer 1

In general uncorrelated does not imply independence.

Your second belief - independence of two R.V. implies independence of functions of the two R.V. is correct.