I am looking to perform a unitary transformation $U$ on a matrix $S$ such that \begin{equation} Q = USU^\dagger \end{equation} is an element-wise non-negative matrix.
I would like to know whether there is a constructive procedure for $U$, and what properties $S$ must have for such a transformation to be possible. For instance, I know that if $\operatorname{Tr}(S)<0$, then such a transformation does not exist.
I am interested in two particular cases:
(a) when $S$ is symmetric and unitary,
(b) when $S$ is diagonal (but non-negative).
Note: I have so far figured out that if $S$ is symmetric, unitary and has $\operatorname{Tr}(S)=0$, then $Q$ must be a complete permutation matrix. But even in this case, is there a way to find whether $Q$ exists or not or how to find it.