Constructions of complex equiangular lines from mutually unbiased bases
Jedwab, Jonathan · Wiebe, Amy
Original · EN
A set of vectors of equal norm in Cᵈ 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 Cᵈ 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 Cᵈ to obtain d² equiangular lines in Cᵈ, (2) using a set of d MUBs in Cᵈ to build (2d)² equiangular lines in C²ᵈ, (3) combining two copies of a set of d MUBs in Cᵈ to build (2d)² equiangular lines in C²ᵈ. 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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