Let $X_1, X_2,\cdots,X_m$ be identical and marginally $Bern(p=0.5)$ random variables. There is no restriction on the joint distribution of $X_1, X_2,\cdots,X_m$.
The entropy $H(X_1, X_2,\cdots,X_m)$ is maximized (over all possible joint distributions) when $X_i's$ are independent. This can be proved by expanding the entropy term using chain rule
Is the entropy of their sum, $S=X_1+X_2+\cdots+X_m$ also maximized when they are independent?