Constructions of complex equiangular lines from mutually unbiased bases

TL;DR

This paper explores three new methods to construct large sets of equiangular lines in complex spaces using mutually unbiased bases, advancing understanding of the maximum possible sizes in various dimensions.

Contribution

It introduces three novel constructions of equiangular lines from mutually unbiased bases, providing specific dimension conditions for maximum size sets.

Findings

Constructs of $d^2$ equiangular lines in $ ext{C}^d$ from $d$ MUBs.

Creates $(2d)^2$ equiangular lines in $ ext{C}^{2d}$ from $d$ MUBs.

Builds $(2d)^2$ equiangular lines in $ ext{C}^{2d}$ by combining two sets of $d$ MUBs.

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 $d \geq 2$. 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.