Some remarks on Heisenberg frames and sets of equiangular lines

Len Bos and Shayne Waldron


We consider the long standing problem of constructing $d^2$ equiangular lines in $\Cd$, i.e., finding a set of $d^2$ unit vectors $(\phi_j)$ in $\Cd$ with $$ |\inpro{\phi_j,\phi_k}|={1\over\sqrt{d+1}}, \qquad j\ne k. $$ Such `equally spaced configurations' have appeared in various guises, e.g., as complex spherical $2$--designs, equiangular tight frames, isometric embeddings $\ell_2(d)\to\ell_4(d^2)$, and most recently as SICPOVMs in quantum measurement theory. Analytic solutions are known only for $d=2,3,4,8$. Recently, numerical solutions which are the orbit of a discrete Heisenberg group $H$ have been constructed for $d\le 45$. We call these Heisenberg frames.

In this paper we study the normaliser of $H$, which we view as a group of symmetries of the equations that determine a Heisenberg frame. This allows us to simplify the equations for a Heisenberg frame. From these simplified equations we are able construct analytic solutions for $d=5,7$, and make conjectures about the form of a solution when $d$ is odd. Most notably, it appears that solutions for $d$ odd are eigenvectors of some element in the normaliser which has (scalar) order $3$.

Keywords: complex spherical 2-design, equiangular lines, equiangular tight frame, Grassmannian frame, Heisenberg frame, isometric embeddings, discrete Heisenberg group modulo d, SICPOVM (symmetric informationally--complete positive operator--valued measure)

Math Review Classification: Primary 05B30, 42C15, 65D30, 81P15; Secondary 51F25, 81R05

Length: 27 pages

Last Updated: 25 June 2007