Consider two identically distributed independent random vectors $X$ and $Y$ of dimension $n$ (assume $n$ is large). Both $X$ and $Y$ have only non-negative integer entries less than or equal to $n$ and $H(X)= H(Y) = n$.
What is the maximum possible value of $H(X + Y)$?
In particular, how close can it get to $2n$?
My working so far
If $X$ and $Y$ are random $0/1$ vectors then $H(X+Y) = 3n/2$ I believe. This is the highest value for $H(X+Y)$ I have managed to find so far.
There is a well known bound on $H(X+Y)$ i.e., $\max{\{H(X),H(Y)\}}\leq H(X+Y) \leq H(X)+H(Y)$. Hence the maximum possible value of $H(X+Y)$ is $H(X)+H(Y)$. Equality holds iff $X=Y$. Hence $H(X+Y)\leq 2n$, and it can come close to $2n$ by picking same value for $X$ and $Y$ always (to be more precise $X=Y$ almost surely).
For proof of above bound, one can refer to http://www2.isye.gatech.edu/~yxie77/ece587/SumRV.pdf