Hjelmslev Geometry of Mutually Unbiased Bases

TL;DR

This paper explores the combinatorial structure of mutually unbiased bases in quantum Hilbert spaces using Hjelmslev geometry, revealing a geometric model over Galois rings that mimics their properties.

Contribution

It introduces a novel geometric framework using Hjelmslev planes over Galois rings to model the properties of mutually unbiased bases in quantum systems.

Findings

Mutually unbiased bases correspond to points on a conic in Hjelmslev geometry.

The number of MUBs matches the count of disjoint neighbour classes on the conic.

The geometric model captures the combinatorial properties of MUBs in prime power dimensions.

Abstract

The basic combinatorial properties of a complete set of mutually unbiased bases (MUBs) of a q-dimensional Hilbert space H\_q, q = p^r with p being a prime and r a positive integer, are shown to be qualitatively mimicked by the configuration of points lying on a proper conic in a projective Hjelmslev plane defined over a Galois ring of characteristic p^2 and rank r. The q vectors of a basis of H\_q correspond to the q points of a (so-called) neighbour class and the q+1 MUBs answer to the total number of (pairwise disjoint) neighbour classes on the conic.