A note on tight projective 2-designs

TL;DR

This paper explores tight projective 2-designs across complex, finite field, and quaternionic settings, establishing new characterizations, constructions, and connections to quantum channels and fusion frames.

Contribution

It introduces a unified approach to understanding tight projective 2-designs in multiple mathematical frameworks, linking quantum information, finite geometry, and frame theory.

Findings

Quantum entanglement breaking rank equals the smallest weighted 2-design size.

Characterization and construction methods for finite field projective 2-designs.

Tight projective 2-designs in quaternionic setting relate to equi-isoclinic tight fusion frames.

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.