Finite Linear Spaces, Plane Geometries, Hilbert spaces and Finite Phase Space

TL;DR

This paper explores the connection between finite plane geometry and finite-dimensional Hilbert spaces, introducing line operators that form a basis for representing quantum states and operators in phase space.

Contribution

It introduces a geometric framework using line operators based on finite plane geometry to represent Hilbert space operators in phase space, with detailed analysis of their properties.

Findings

Line operators form a complete orthogonal basis for Hilbert space operators

Geometric point-line relations underpin the operator representations

Phase space representation is achieved through these geometric constructs

Abstract

Finite plane geometry is associated with finite dimensional Hilbert space. The association allows mapping of q-number Hilbert space observables to the c-number formalism of quantum mechanics in phase space. The mapped entities reflect geometrically based line-point interrelation. Particularly simple formulas are involved when use is made of mutually unbiased bases (MUB) representations for the Hilbert space entries. The geometry specifies a point-line interrelation. Thus underpinning d-dimensional Hilbert space operators (resp. states) with geometrical points leads to operators termed "line operators" underpinned by the geometrical lines. These "line operators", $\hat{L}_j;$ (j designates the line) form a complete orthogonal basis for Hilbert space operators. The representation of Hilbert space operators in terms of these operators form the phase space representation of the d-dimensional Hilbert space. The "line operators" (resp. "line states") are studied in detail. The paper aims at self sufficiency and to this end all relevant notions are explained herewith.