# Some remarks on Heisenberg frames and sets of equiangular lines

## Len Bos and Shayne Waldron

## Abstract:

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

## Availability:

- pdf
- pdf
(New Zealand Journal of Mathematics)