Constructions of complex equiangular lines from mutually unbiased bases

Abstract

A set of vectors of equal norm in $$\mathbb {C}^d$$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 $$d^2$$d2, and it is conjectured that sets of this maximum size exist in $$\mathbb {C}^d$$Cd for every $$d \ge 2$$dź2. We take a combinatorial approach to this conjecture, using mutually unbiased bases (MUBs) in the following three constructions of equiangular lines:(1)adapting a set of $$d$$d MUBs in $$\mathbb {C}^d$$Cd to obtain $$d^2$$d2 equiangular lines in $$\mathbb {C}^d$$Cd,(2)using a set of $$d$$d MUBs in $$\mathbb {C}^d$$Cd to build $$(2d)^2$$(2d)2 equiangular lines in $$\mathbb {C}^{2d}$$C2d,(3)combining two copies of a set of $$d$$d MUBs in $$\mathbb {C}^d$$Cd to build $$(2d)^2$$(2d)2 equiangular lines in $$\mathbb {C}^{2d}$$C2d. For each construction, we give the dimensions $$d$$d for which we currently know that the construction produces a maximum-sized set of equiangular lines.