% Discrete Math Day Problem session
% Problem submission -- sample format (wjm, 8/31/2015)
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\noindent Discrete Math Day \\
\noindent Problem Session Contribution \\
\noindent William J.~Martin \\
\noindent September 12, 2015
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{\bf Equiangular Lines in Complex Space} \\
{\sc Bill Martin (wpi)}
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\noindent {\bf Problem:} Find 306 lines in $\cx^{18}$ such that any two
form the same angle. Equivalently, for $d=18$, find $d^2-d$ unit vectors
$\{ \v_i \mid 1\le i \le d(d-1) \}$ such that, for some real $\alpha$,
$ | \v_i \v_j^* | = \alpha $ for all $i\neq j$.
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\noindent {\bf Discussion:} A set of $d^2$ equiangular lines in $\cx^d$ is
called a ``symmetric informationally complete positive operator-valued measure''
(SIC-POVM, for short) and these are of interest in quantum information theory,
e.g., for quantum state tomography. In order to attain $d^2$ vectors, it is easy
to prove that $\alpha = 1/\sqrt{d+1}$ is required.
Gerhard Zauner conjectured in 1999 that these exist for all $d$ and conjectured
a method for constructing them. Exact solutions have been found by Gr\"{o}bner
basis methods for $2\le d\le 17$, and $d=19, 24, 35, 48$. High-precision numerical approximations
have been found for $d\le 121$. So $d=18$ is the smallest open case for an
exact solution. Here, we instead ask for $d^2-d$ equiangular lines in the hopes that (a) the problem may be more tractible and (b) a solution to the easier problem may provide insight into the SIC-POVM problem.
\noindent {\bf References:} A survey article can be found here: {\tt http://arxiv.org/abs/0910.5784}
and a recent talk which provides an update can be found here: \\
{\tt http://users.wpi.edu/~martin/MEETINGS/LINESTALKS/Appleby.pdf}
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\noindent {\bf Contact Information:} Bill Martin, Worcester Polytechnic Institute,
{\tt martin@wpi.edu}
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