Some remarks on Heisenberg frames and sets of equiangular lines
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چکیده
We consider the long standing problem of constructing d equiangular lines in C, i.e., finding a set of d unit vectors (φj) in C d with |〈φj , φk〉| = 1 √ d + 1 , j 6= k. Such ‘equally spaced configurations’ have appeared in various guises, e.g., as complex spherical 2–designs, equiangular tight frames, isometric embeddings `2(d) → `4(d), 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 ≤ 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.
منابع مشابه
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تاریخ انتشار 2006