2- and 3-Covariant Equiangular Tight Frames

Abstract

Equiangular tight frames (ETFs) are configurations of vectors which are optimally geometrically spread apart and provide resolutions of the identity. Many known constructions of ETFs are group covariant, meaning they result from the action of a group on a vector, like all known constructions of symmetric, informationally complete, positive operator-valued measures. In this short article, some results characterizing the transitivity of the symmetry groups of ETFs will be presented as well as a proof that an infinite class of so-called Gabor-Steiner ETFs are roux lines, where roux lines are a generalization of doubly transitive lines.

Optimal arrangements of classical and quantum states with limited purity

“…In the preceding results, we showed that Gabor–Steiner equiangular tight frames over any type of finite abelian group have rich binders. It is proven in that for odd prime has a high level of symmetry. We believe that could provide a set of examples of equiangular tight frames with a variety of symmetry groups for more general choices of the vector .…”

Testing isomorphism between tuples of subspaces

“…Some of the ideas in the paper may have interesting applications elsewhere. For example, there has been a lot of work to develop symmetric arrangements of points in the Grassmannian [62,11,8,65,61,10,35,36,37,45,7,42,38,29,19]. What are the projection and cross Gramian algebras of these arrangements?…”

“…First, it is natural to consider highly symmetric arrangements of points. Such arrangements were extensively studied in [62,11,8,65,61,10] in the context of designs, and later, symmetry was used to facilitate the search for optimal codes [35,36,37,45,7,42,38]. In many cases, the symmetries that underly optimal codes can be abstracted to weaker combinatorial structures that produce additional codes.…”