Radon Transform in Finite Dimensional Hilbert Space

TL;DR

This paper introduces a novel finite-dimensional Radon transform framework in Hilbert spaces, leveraging finite geometry and mutual unbiased bases to enhance understanding of quantum operators and phase space representations.

Contribution

It presents a new approach to finite-dimensional quantum analysis using finite geometry, extending the use of MUBs and formulating a Radon transform with inverse in this setting.

Findings

Finite geometry underpins the analysis of quantum operators.

Radon transform and inverse are formulated in finite dimensions.

Mutual unbiased bases are central to the approach.

Abstract

Novel analysis of finite dimensional Hilbert space is outlined. The approach bypasses general, inherent, difficulties present in handling angular variables in finite dimensional problems: The finite dimensional, d, Hilbert space operators are underpinned with finite geometry which provide intuitive perspective to the physical operators. The analysis emphasizes a central role for projectors of mutual unbiased bases (MUB) states, extending thereby their use in finite dimensional quantum mechanics studies. Interrelation among the Hilbert space operators revealed via their (finite) dual affine plane geometry (DAPG) underpinning are displayed and utilized in formulating the finite dimensional ubiquitous Radon transformation and its inverse illustrating phase space-like physics encoded in lines and points of the geometry. The finite geometry required for our study is outlined.