Some remarks on Heisenberg frames and sets of equiangular lines. Bos, L. & Waldron, S. New Zealand J. Math., 36:113--137, 2007. abstract bibtex We consider the long standing problem of constructing $d^2$ equiangular lines in $C^d$, i.e., finding a set of $d^2$ unit vectors ($φ_j$) in $C^d$ with $$|\langle\phi_j,\phi_kångle|=\frac1\sqrtd+1, \qquad j\ne k.$$ 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 $d = 2,3,4,5,6,8$ and $d = 7,19$ (Appleby 2005). Recently, numerical solutions which are the orbit of a discrete Heisenberg group $H$ have been constructed for $d\le 45$. 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, e.g., for $d$ odd we have ${1\over8}d^2+{7\over 8}$ 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.
@article {MR2455575,
AUTHOR = {Bos, Len and Waldron, Shayne},
TITLE = {Some remarks on {H}eisenberg frames and sets of equiangular
lines},
JOURNAL = {New Zealand J. Math.},
FJOURNAL = {New Zealand Journal of Mathematics},
VOLUME = {36},
YEAR = {2007},
PAGES = {113--137},
ISSN = {1171-6096},
MRCLASS = {05B30 (42C15 81P15)},
MRNUMBER = {2455575 (2009i:05051)},
ABSTRACT = {We consider the long standing problem of constructing $d^2$ equiangular lines in $C^d$, i.e., finding a set of $d^2$ unit vectors ($φ_j$) in $C^d$ with
$$|{\langle\phi_j,\phi_k\rangle}|={\frac{1}{\sqrt{d+1}}}, \qquad j\ne k.$$
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 $d = 2,3,4,5,6,8$ and $d = 7,19$ (Appleby 2005). Recently, numerical solutions which are the orbit of a discrete Heisenberg group $H$ have been constructed for $d\le 45$. 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, e.g., for $d$ odd we have ${1\over8}d^2+{7\over 8}$ 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. }
}
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