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How to find the Least Entropy Solution From the Product of Known Entropy

Let’s say $H(X)=2\cdot lg(2)$ and $H(Y)=1\cdot lg(2)$ where $X$ and $Y$ are two random variables. How can I find the Least Entropy Solution (L.E.S) of the product $H(X\cdot Y)$.

A:

Asking how to find the LE solution of a product is essentially asking how to factorize a number.
In the case of $H(X \cdot Y)= 2 \cdot 1\cdot lg(2) = 2 \cdot lg(2) = 2lg(2) = 2 \cdot H(X)$
It can be seen that the relative entropy of the product of the two random variables is equal to the total entropies of the original random variables.
So just as it is the case with division, the product of two (independent) random variables can be written as the product of the factors.

Another useful relation is that for two random variables \$H(XY)

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