Finite linear spaces, plane geometries, Hilbert spaces and finite phase space

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 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. Examples for the use of the “line operators” in mapping (finite dimensional) Hilbert space operators onto finite dimensional phase space functions are considered. These include finite dimensional Wigner function and Radon transform and a geometrical interpretation for the involvement of parity in the mappings of Hilbert space onto phase space. Two d-dimensional particles product states are underpinned with geometrical points. The states, \(|L_j\rangle \) underpinned with the corresponding geometrical lines are maximally entangled states (MES). These “line states” provide a complete \(d^2\) dimensional orthogonal MES basis for for the two d-dimensional particles. The complete \(d^2\) dimensional MES i.e. the “line states” are shown to provide a transparent geometrical interpretation to the so-called Mean King Problem and its variant. 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.