I am trying to make sense of the following condition:
Let $w_1, \dots, w_m \in \mathbb{C}^d$ with $\|w_i\| \le 1$ and $\sum_{i = 1}^m \, |\langle u, w_i \rangle |^2 = n$ for some $n \in \mathbb{R}$ and any unit vector $u$.
Geometrically speaking, what does this condition tell me about my $w_i$ vectors?
The first condition $\|w_i\|\leq 1$ means that all of your $w_i$ lie on or inside the unit ball. Easy enough.
The second is basically saying, "pick a direction, and take the component of the vectors that lie in that direction. Add up the squares of these components, and they have to add up to $n$."
To me that suggests that the $w_i$ are somewhat evenly distributed, direction wise.