Let $H$ be a $4 \times 2$ matrix (with entries drawn from a unit variance zero mean complex normal distribution, however I'm not sure this detail is necessary).
Let $h_1$ denote the first column of $H$ and $h_2$ the second. Then let $c_1 = 2*Re\{h_1^H h_2\}$ and $c_2 = 2*Im\{h_1^H h_2\}$ Finally let $\theta = k\pi + tan^{-1}(\frac{-c_1 - c_2}{c_1 - c_2})$ where $k$ is an integer chosen to keep $ 0 \leq \theta \leq \pi$.
Then we have the following function:
$f(H) = 4\frac{||h_1||^2 ||h_2||^2}{||h_1||^2 + ||h_2||^2} - 4Re\{\sqrt{\frac{||h_1||^2 ||h_2||^2}{||h_1||^2 + ||h_2||^2}} h_1^H h_2 e^{j \theta}\}$
The goal is, given a finite set of $H$ realizations, find the one that maximizes this function. By this I mean, find characteristics that let you choose the best $H$ without having to plug in each $H$ and evaluate the function for every possible one (realistically these characteristics would likely be suboptimal, and ideally these characteristics would be lower complexity than going and doing this full evaluation).
My thoughts on the approach was to think of the function taking a vector $h = [h_1^T, h_2^T]$ e.g. the function is $\mathbb{C}^8 \rightarrow \mathbb{R}$ and then compute the gradient to find critical points and hopefully identify a trend. I'm not really sure that this will work, would appreciate any push in the right direction.