Maximal Sets of Equiangular Lines

TL;DR

This paper explores the problem of determining the maximum number of equiangular lines in various dimensions, highlighting its connections to quantum information, octonions, and number theory, and emphasizing its unresolved nature.

Contribution

The paper provides an overview of the problem of maximal equiangular lines, discussing its significance and the challenges involved in solving it in different spaces.

Findings

The problem is easy to state but remains unresolved.

Connections to quantum information theory and octonions are highlighted.

The problem has a classic charm due to its simplicity and difficulty.

Abstract

I introduce the problem of finding maximal sets of equiangular lines, in both its real and complex versions, attempting to write the treatment that I would have wanted when I first encountered the subject. Equiangular lines intersect in the overlap region of quantum information theory, the octonions and Hilbert's twelfth problem. The question of how many equiangular lines can fit into a space of a given dimension is easy to pose -- a high-school student can grasp it -- yet it is hard to answer, being as yet unresolved. This contrast of ease and difficulty gives the problem a classic charm.