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 {ai} and {bj} are called mutually unbiased if all inner products ⟨ai∣bj⟩ have the same modulus 1∕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 such N+1 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.