Characterizing maximal families of mutually unbiased bases

We show that maximal families of mutually unbiased bases are characterized in all dimensions by partitioned unitary error bases, up to a choice of a family of Hadamards. Furthermore, we give a new construction of partitioned unitary error bases, and thus maximal families of mutually unbiased bases, from a finite field, which is simpler and more direct than previous proposals. We introduce new tensor diagrammatic characterizations of maximal families of mutually unbiased bases, partitioned unitary error bases, and finite fields as algebraic structures defined over Hilbert spaces.