A note on tight projective 2-designs
Abstract
We study tight projective 2-designs in three different settings. In the complex setting, Zauner's conjecture predicts the existence of a tight projective 2-design in every dimension. Pandey, Paulsen, Prakash, and Rahaman recently proposed an approach to make quantitative progress on this conjecture in terms of the entanglement breaking rank of a certain quantum channel. We show that this quantity is equal to the size of the smallest weighted projective 2-design. Next, in the finite field setting, we introduce a notion of projective 2-designs, we characterize when such projective 2-designs are tight, and we provide a construction of such objects. Finally, in the quaternionic setting, we show that every tight projective 2-design for H^d determines an equi-isoclinic tight fusion frame of d(2d-1) subspaces of R^d(2d+1) of dimension 3.
Keywords
Cite
@article{arxiv.2101.11756,
title = {A note on tight projective 2-designs},
author = {Joseph W. Iverson and Emily J. King and Dustin G. Mixon},
journal= {arXiv preprint arXiv:2101.11756},
year = {2021}
}