Constructions of complex equiangular lines from mutually unbiased bases

A set of vectors of equal norm in 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 , and it is conjectured that sets of this maximum size exist in for every . We take a combinatorial approach to this conjecture, using mutually unbiased bases (MUBs) in the following three constructions of equiangular lines: adapting a set of MUBs in to obtain equiangular lines in , using a set of MUBs in to build equiangular lines in , combining two copies of a set of MUBs in to build equiangular lines in . For each construction, we give the dimensions d for which we currently know that the construction produces a maximum-sized set of equiangular lines.