I am currently trying to understand when the following system of equations has a solution: $$x^t U_i^\dagger U_j x = c_{ij}$$ Where the $U_i$ are $n$ by $n$ unitary matrices, $x \in \mathbb{C}^n$, $c_{ij}$ is a $k$ by $k$ matrix and $i$, $j$ are indices that run from 1 to $k$.
In particular, I would like to know what are some nontrivial necessary and/or sufficient conditions on $c_{ij}$ so that a solution exists. For example, a necessary condition would be $c_{ii}=1$.
I've looked at a lot of questions and answers on this site, but they all seem to relate to specific instances (such as real variables) while I'm looking for either a general answer, or an answer specific to the unitary case. The only one that comes close is this answer, but it's very vague.
Any help would be appreciated.
Potential simplifying assumptions/other constraints
- I am ok with assuming that the eigenvalues of the $U_{ij}$ are $\pm 1$
- If it's necessary, I can assume that $c_{ij} \in \{0,1\}$
- $x$ should have unit norm