We study tight projective 2-designs in three different settings. In the complex setting, Zauners 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.
We investigate equiangular lines in finite orthogonal geometries, focusing specifically on equiangular tight frames (ETFs). In parallel with the known correspondence between real ETFs and strongly regular graphs (SRGs) that satisfy certain parameter
constraints, we prove that ETFs in finite orthogonal geometries are closely aligned with a modular generalization of SRGs. The constraints in our finite field setting are weaker, and all but~18 known SRG parameters on $v leq 1300$ vertices satisfy at least one of them. Applying our results to triangular graphs, we deduce that Gerzons bound is attained in finite orthogonal geometries of infinitely many dimensions. We also demonstrate connections with real ETFs, and derive necessary conditions for ETFs in finite orthogonal geometries. As an application, we show that Gerzons bound cannot be attained in a finite orthogonal geometry of dimension~5.
We introduce the study of frames and equiangular lines in classical geometries over finite fields. After developing the basic theory, we give several examples and demonstrate finite field analogs of equiangular tight frames (ETFs) produced by modular
difference sets, and by translation and modulation operators. Using the latter, we prove that Gerzons bound is attained in each unitary geometry of dimension $d = 2^{2l+1}$ over the field $mathbb{F}_{3^2}$. We also investigate interactions between complex ETFs and those in finite unitary geometries, and we show that every complex ETF implies the existence of ETFs with the same size over infinitely many finite fields.
An equiangular tight frame (ETF) is a sequence of vectors in a Hilbert space that achieves equality in the Welch bound and so has minimal coherence. More generally, an equichordal tight fusion frame (ECTFF) is a sequence of equi-dimensional subspaces
of a Hilbert space that achieves equality in Conway, Hardin and Sloanes simplex bound. Every ECTFF is a type of optimal Grassmannian code, that is, an optimal packing of equi-dimensional subspaces of a Hilbert space. We construct ECTFFs by exploiting new relationships between known ETFs. Harmonic ETFs equate to difference sets for finite abelian groups. We say that a difference set for such a group is paired with a difference set for its Pontryagin dual when the corresponding subsequence of its harmonic ETF happens to be an ETF for its span. We show that every such pair yields an ECTFF. We moreover construct an infinite family of paired difference sets using quadratic forms over the field of two elements. Together this yields two infinite families of real ECTFFs.
