Geometrical Underpinning of Finite Dimensional Hilbert space

TL;DR

This paper explores the geometric structure underlying finite-dimensional Hilbert spaces, emphasizing the role of mutual unbiased bases and their relation to finite affine geometries, providing a new perspective on operator interrelations.

Contribution

It introduces a geometric framework based on finite affine and dual affine plane geometries to understand operators and MUB states in finite Hilbert spaces, offering novel insights.

Findings

Operators are interconnected through finite geometric structures.

Mutual unbiased bases are central to the geometric interpretation.

Finite geometries facilitate new interpretations of Hilbert space operators.

Abstract

Finite geometry is employed to underpin operators in finite, d, dimensional Hilbert space. The central role of mutual unbiased bases (MUB) states projectors is exhibited. Interrelation among operators in Hilbert space, revealed through their (finite) dual affine plane geometry (DAPG) underpinning is studied. Transcription to (finite) affine plane geometry (APG) is given and utilized for their interpretation.