Finite Geometry and the Radon Transform

TL;DR

This paper explores how finite geometry, specifically dual affine plane geometry, underpins operators, quasi distributions, and the Radon transform in finite-dimensional Hilbert spaces, drawing analogies with continuous cases.

Contribution

It introduces a finite geometric framework for the Radon transform and quasi distributions using dual affine plane geometry and MUB projectors.

Findings

Defined finite Radon transform using DAPG

Established analogy with continuous Hilbert space Radon transform

Linked MUB projectors with phase space mappings

Abstract

Finite Geometry is used to underpin operators acting in finite, d, dimensional Hilbert space. Quasi distribution and Radon transform underpinned with finite dual affine plane geometry (DAPG) are defined in analogy with the continuous ($d \rightarrow \infty$) Hilbert space case. An essntial role in these definitions play the projectors of states of mutual unbiased bases (MUB) and their Wigner function-like mapping onto the generalized phase space that lines and points of DAPG constitutes.