The Weyl basis (also known as Weyl-Heisenberg) is an orthonormal, unitary, and non-Hermitian basis for the Hilbert space of dimension $d$. The basis elements are given by $$ U_{ab} = \sqrt{\omega^{ab}}X^aZ^b, $$ where $X,Z$ are the shift and clock matrices and $\omega = e^{i2\pi/d}$. I'm wondering whether for all $d$ there exists a pure state $|\psi\rangle$ such that $|\langle\psi| U_{ab}|\psi\rangle|$ is constant (except of course for $U_{00}$).
Expanding the hypothetical state in this basis, $$|\psi\rangle\langle\psi| = \frac1d \sum_{ab} \alpha_{a,b} U_{ab},$$ we see that demanding it to be Hermitian translates to $\alpha_{a,b}\alpha_{-a,-b} = 1$ (note that these coefficients are in general complex), have trace one results in $\alpha_{0,0} = 1$, and have trace squared equal one results in $|\alpha_{a,b}| = 1/\sqrt{d+1}$ for all other $a,b$.
Simply setting $\alpha_{a,b} = 1/\sqrt{d+1}$ does give me the desired state for $d=2$ and $d=3$, but doesn't work for higher dimensions, negative eigenvalues show up. It's easy enough to find the state numerically for higher $d$, so I guess it always exists, but I can't find a good method to calculate the coefficients analytically. Also, it seems a natural question to ask, but I couldn't find anything about it in the literature.
