Say we have a function $f:\mathbb{Z}_2^n \to \mathbb{R}$, such that $\sum _{x\in \mathbb{Z}_2^n} f(x)^2 = 1$ (so we can think of $\{ f(x)^2\} _{x\in \mathbb{Z}_2^n}$ as a distribution). It is natural to define the entropy of such function $f$ as follows: $$H(f) = -\sum _{x \in \mathbb{Z}_2^n} f(x)^2 \log \left( f(x)^2 \right) .$$

Now, consider the convolution of $f$ with itself: $$ [ f*f](x) = \sum_{y \in \mathbb{Z}_2^n}f(y)f(x+y) .$$ (Since we are dealing with $\mathbb{Z}_2^n$, then $x+y=x-y$)

Is it possible to upper bound the entropy of $f*f$ (normalized in its $L_2$-norm, in order for it to be a distribution)? Formally, is there any constant $C$ such that $$ H \left( \frac{f*f}{\|f*f\|_2} \right) \le C \cdot H(f)$$

Note: This question was later posted to cstheory.stackexchange.com, where it was answered by Colin McQuillan.