There is no generalization of known formulas for mutually unbiased bases

In a quantum system having a finite number $N$ of orthogonal states, two orthonormal bases $\{a_i\}$ and $\{b_j\}$ are called mutually unbiased if all inner products $$ have the same modulus $1/\sqrt{N}$. This concept appears in several quantum information problems. The number of pairwise mutually unbiased bases is at most $N+1$ and various constructions of $N+1$ such bases have been found when $N$ is a power of a prime number. We study families of formulas that generalize these constructions to arbitrary dimensions using finite rings.We then prove that there exists a set of $N+1$ mutually unbiased bases described by such formulas, if and only if $N$ is a power of a prime number.