On the monomiality of nice error bases

Unitary error bases generalize the Pauli matrices to higher dimensional systems. Two basic constructions of unitary error bases are known: An algebraic construction by Knill that yields nice error bases, and a combinatorial construction by Werner that yields shift-and-multiply bases. An open problem posed by Schlingemann and Werner relates these two constructions and asks whether each nice error basis is equivalent to a shift-and-multiply basis. We solve this problem and show that the answer is negative. However, we find that nice error bases have more structure than one can anticipate from their definition. In particular, we show that nice error bases can be written in a form in which at least half of the matrix entries are 0.