Qudits of composite dimension, mutually unbiased bases and projective ring geometry

TL;DR

This paper explores the structure of qudits in composite dimensions using symplectic modules and projective ring geometry, revealing how commuting operators and mutually unbiased bases relate to algebraic structures over rings.

Contribution

It establishes a novel connection between the geometry of qudit operators and projective lines over rings, extending understanding of mutually unbiased bases in composite dimensions.

Findings

Reveals the structure of maximal commuting sets as projective lines over rings.

Shows the incidence relations correspond to algebraic structures over specific rings.

Provides geometric interpretation for mutually unbiased bases in composite dimensions.

Abstract

The $d^2$ Pauli operators attached to a composite qudit in dimension $d$ may be mapped to the vectors of the symplectic module $\mathcal{Z}_d^{2}$ ($\mathcal{Z}_d$ the modular ring). As a result, perpendicular vectors correspond to commuting operators, a free cyclic submodule to a maximal commuting set, and disjoint such sets to mutually unbiased bases. For dimensions $d=6,~10,~15,~12$, and 18, the fine structure and the incidence between maximal commuting sets is found to reproduce the projective line over the rings $\mathcal{Z}_{6}$, $\mathcal{Z}_{10}$, $\mathcal{Z}_{15}$, $\mathcal{Z}_6 \times \mathbf{F}_4$ and $\mathcal{Z}_6 \times \mathcal{Z}_3$, respectively.