Constructions of complex equiangular lines from mutually unbiased bases

2014/08/21 by Jedwab, Jonathan, Wiebe, Amy
#Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.1408.5169

Abstract

A set of vectors of equal norm in ℂ^d represents equiangular lines if the magnitudes of the Hermitian inner product of every pair of distinct vectors in the set are equal. The maximum size of such a set is d², and it is conjectured that sets of this maximum size exist in ℂ^d for every d ≥ 2. We take a combinatorial approach to this conjecture, using mutually unbiased bases (MUBs) in the following 3 constructions of equiangular lines: (1) adapting a set of d MUBs in ℂ^d to obtain d² equiangular lines in ℂ^d, (2) using a set of d MUBs in ℂ^d to build (2d)² equiangular lines in ℂ^2d, (3) combining two copies of a set of d MUBs in ℂ^d to build (2d)² equiangular lines in ℂ^2d. For each construction, we give the dimensions d for which we currently know that the construction produces a maximum-sized set of equiangular lines.

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