Mutually unbiased bases as submodules and subspaces

TL;DR

This paper explores the relationship between mutually unbiased bases (MUBs) and advanced finite geometries, proposing new geometric structures that underpin various MUB constructions, with implications for cryptography and communications.

Contribution

It introduces a novel connection between MUBs and higher-dimensional projective and Hjelmslev geometries, expanding the geometric understanding of MUBs beyond classical planes.

Findings

MUBs are linked to higher-dimensional projective geometries.

The proposed geometric structures are present in multiple MUB constructions.

This connection offers new insights into the structure of MUBs.

Abstract

Mutually unbiased bases (MUBs) have been used in several cryptographic and communications applications. There has been much speculation regarding connections between MUBs and finite geometries. Most of which has focused on a connection with projective and affine planes. We propose a connection with higher dimensional projective geometries and projective Hjelmslev geometries. We show that this proposed geometric structure is present in several constructions of MUBs.