Mutually unbiased bases as minimal Clifford covariant 2-designs

Huangjun Zhu

Published 2015-05-05Version 1

Mutually unbiased bases (MUB) are interesting for various reasons. The canonical MUB constructed by Ivanovi\'c as well as Wootters and Fields have attracted the most attention. Nevertheless, little is known about anything that is unique to this MUB. We show that the canonical MUB in any prime power dimension is uniquely determined by an extremal orbit of the (restricted) Clifford group except in dimension 3, in which case the trophy is taken by a special symmetric informationally complete measurement (SIC), known as the Hesse SIC. Here the extremal orbit is the orbit with the smallest number of pure states. Quite surprisingly, this characterization does not rely on any concept that is even remotely related to bases or unbiasedness. As a corollary, the canonical MUB is the unique minimal 2-design covariant with respect to the Clifford group except in dimension 3. In addition, these MUB provide an infinite family of highly symmetric frames and positive-operator-valued measures (POVMs), which are of independent interest.