Some Remarks on Heisenberg Frames and Sets of Equiangular Lines
From NZJM
New Zealand Journal of Mathematics
Vol. 36, (2007), Pages 113-137
Len Bos
Department of Mathematics and Statistics
University of Calgary
Calgary, Alberta
CANADA T2N1N4
Shayne Waldron
Department of Mathematics
University of Auckland
Private Bag 92019
Auckland
NEW ZEALAND
mailto:waldron@math.auckland.ac.nz
Abstract We consider the long standing problem of constructing d2 equiangular lines in Cd, i.e., finding a set of d2 unit vectors (φj) in Cd with
Such `equally spaced configurations' have appeared in various guises, e.g.,
as complex spherical 2-designs, equiangular tight frames,
isometric embeddings 
, and
most recently as SICPOVMs in quantum measurement theory.
Analytic solutions are known only for d = 2,3,4,5,6,8
and d = 7,19 (Appleby 2005).
Recently, numerical solutions which are the orbit of a discrete Heisenberg group H
have been constructed for 
.
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, e.g.,
for d odd we have
$ real equations in the d coordinates of v
and their complex conjugates.
From these simplified equations we are able construct analytic solutions
for d = 5,7, and make conjectures about the form of a solution.
It is hoped that a general solution will come from such a
simplified set of equations.
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)
Classification (MSC2000) Primary: 05B30, 42C15, 65D30, 81P15
