All complex equiangular tight frames in dimension 3

TL;DR

This paper fully classifies all complex equiangular tight frames in three-dimensional space, revealing new algebraic properties and nonexistence results for certain low-dimensional cases using computational algebra techniques.

Contribution

It provides the first complete algebraic classification of complex ETFs in C^3 and introduces new algebraic features of their Gram matrices.

Findings

Complete classification of complex ETFs in C^3

Nonexistence of certain low-dimensional complex ETFs

Identification of new algebraic properties of Gram matrices

Abstract

In this paper we describe some new algebraic features of the Gram matrices of complex Equiangular Tight Frames (ETF). This lead on the one hand to the nonexistence of several low dimensional complex ETFs; and on the other hand to the full algebraic classification of all complex ETFs in C^3. We use computer aided methods, in particular, Groebner basis computations to obtain these results.