Mutually unbiased bases as minimal Clifford covariant 2-designs

TL;DR

This paper characterizes the canonical mutually unbiased bases (MUB) in prime power dimensions as unique extremal orbits of the Clifford group, revealing their special symmetry properties and connections to 2-designs and SICs.

Contribution

It demonstrates that the canonical MUBs are uniquely determined by extremal Clifford orbits, except in dimension 3, where they relate to the Hesse SIC, providing new insights into their structure.

Findings

Canonical MUBs are uniquely determined by Clifford group orbits in prime power dimensions.

In dimension 3, the orbit defines the Hesse SIC, a special symmetric informationally complete measurement.

Canonical MUBs are the minimal 2-designs covariant with respect to the Clifford group, except in dimension 3.

Abstract

Mutually unbiased bases (MUB) are interesting for various reasons. The most attractive example of (a complete set of) MUB is the one constructed by Ivanovi\'c as well as Wootters and Fields, which is referred to as the canonical MUB. 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 orbit defines a special symmetric informationally complete measurement (SIC), known as the Hesse SIC. Here the extremal orbit is the one with the smallest number of pure states. Quite surprisingly, this characterization does not rely on any concept that is 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.