# Codes and Expansions (CodEx) Seminar

## Emily J. King (Colorado State University)

A Potpourri of Projective 2-Designs

Zauner conjectured that for all \(d\) there exists a set of \(d^2\) equiangular vectors in \(\mathbb{C}^d\) (SICs or Gabor equiangular tight frames). Another important open problem concerns the existence of \(d+1\) so-called mutually unbiased bases in \(\mathbb{C}^d\) (maximal sets of MUBs) for any \(d\) not equal to a prime power. Both classes of objects are complex projective \(2\)-designs. In this talk, the basics of (weighted) projective \(2\)-designs will be discussed, including their recent generalization to finite fields. A collection of new results will be presented.

Pandey, Paulsen, Prakash, and Rahaman recently proposed an approach to make quantitative progress on Zauner's Conjecture in terms of the entanglement breaking rank of a certain quantum channel. This quantity is equal to the size of the smallest weighted projective \(2\)-design. A construction of an infinite class of projective \(2\)-designs over finite fields will also be presented. This construction makes use of difference sets. Finally, in the quaternionic setting, every tight projective \(2\)-design for \(\mathbb{H}^d\) determines an equi-isoclinic tight fusion frame of \(d(2d - 1)\) subspaces of \(\mathbb{R}^{d(2d+1)}\) of dimension \(3\).

This is joint work with Joey Iverson and Dustin Mixon. arXiv:2101.11756