I have $3\times 3$ a matrix $A$ defined as
$$A=e^{i\alpha H}, $$
with $H$ a $3\times 3$ random Hermitian matrix, and $\alpha\in[0,\infty]$. I am trying to determine two things: Can we say anything on the expectation of $A$ in terms of the statistics of $H$? What conditions do $H$ need to satisfy so that the expectation of $A$ is diagonal?
Numerically I find that if the elements of $H$ have zero mean then the expectation of $A$ is diagonal. But I would like to know the general conditions for this (and why), and an explicit expression in terms of the statistics of $H$, if it exists!
Here is a possible reason why your expectation is diagonal under uniform or gaussian distribution.
Let $S$ denote the matrix $$S = \begin{bmatrix} 1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & 1 \end{bmatrix}.$$ Then the distribution of $H$ is invariant under conjugation by $S$, so $$\mathbb{E} e^{i\alpha H} = \mathbb{E} e^{i\alpha S^{-1}HS} = \mathbb{E} S^{-1}e^{i\alpha H}S = S^{-1} \mathbb{E} e^{i\alpha H} S.$$ As conjugation by $S$ negates the elements at $(1, 2), (2, 1), (2, 3), (3, 2)$, we conclude that the elements at those points of $\mathbb{E} e^{i\alpha H}$ are zero. Similarly, by studying $$S' = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & -1 \end{bmatrix}$$ We conclude that if your distribution is invariant under conjugation by $S$ and $S'$, then the expectation $\mathbb{E} e^{i\alpha H}$ is diagonal.