Constructions of complex equiangular lines from mutually unbiased bases

A set of vectors of equal norm in C 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 d, and it is conjectured that sets of this maximum size exist in C 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 to obtain d equiangular lines in C, (2) using a set of d MUBs in C to build (2d) equiangular lines in C, (3) combining two copies of a set of d MUBs in C to build (2d) equiangular lines in C. For each construction, we give the dimensions d for which we currently know that the construction produces a maximum-sized set of equiangular lines.