Constructions of complex equiangular lines from mutually unbiased bases

Abstract

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

arXiv comment20 pages, minor revisions

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