A note on tight projective 2‐designs. Issue 12 (5th September 2021)

• Record Type: Journal Article
• Authors: Iverson, Joseph W.; King, Emily J.; Mixon, Dustin G.
• Is Part Of: Journal of combinatorial designs. Volume 29:Issue 12(2021)
• Page Start: 809
• Page End: 832
• Publication Date: 2021-09-05
• Subjects: projective 2‐designs; Combinatorial designs and configurations -- Periodicals; Configurations et schémas combinatoires -- Périodiques; 511.6
• DOI: 10.1002/jcd.21804
• Languages: English
• ISSNs: 1063-8539

Abstract

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 ( 2 d − 1 ) subspaces of R d ( 2 d + 1 ) of dimension 3.