Mutually unbiased bases as minimal Clifford covariant 2-designs
arXiv:1505.01123 · doi:10.1103/PhysRevA.91.060301
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 Ivanović 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.
5.3 pages; published in PRA (rapid communications)
References in corpus (8)
- Evenly distributed unitaries: on the structure of unitary designs
- Tight informationally complete quantum measurements
- Classicality in discrete Wigner functions
- Properties of the extended Clifford group with applications to SIC-POVMs and MUBs
- Tomographic and Lie algebraic significance of generalized symmetric informationally complete measurements
- Pairs of Generators for Matrix Groups. I
- States that "look the same" with respect to every basis in a mutually unbiased set
- Sharply covariant mutually unbiased bases
Cited by in corpus (5)
- Universally Fisher-Symmetric Informationally Complete Measurements
- Projective toric designs, quantum state designs, and mutually unbiased bases
- Cyclic measurements and simplified quantum state tomography
- Uncertainty conservation relations: theory and experiment
- Generalized Numerical Construction of MUBs: A Group Theoretical Investigation