## 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.

## Full Text

### Topics from this Paper

- Finite Dimensional Hilbert Space
- Finite Phase Space
- Maximally Entangled States
- Hilbert Space Operators
- Line Operators + Show 5 more

Create a personalized feed of these topics

Get Started#### Talk to us

Join us for a 30 min session where you can share your feedback and ask us any queries you have

Schedule a call### Similar Papers

- arXiv: Quantum Physics
- Jun 24, 2009

- Journal of Mathematical Physics
- Oct 1, 2017

- arXiv: Category Theory
- Jul 30, 2012

- arXiv: High Energy Physics - Theory
- Sep 18, 1995

- Pure and Applied Mathematics Journal
- May 12, 2015

- arXiv: Quantum Physics
- Feb 10, 2010

- arXiv: Quantum Physics
- Sep 13, 2016

- arXiv: Functional Analysis
- Jul 26, 2010

- Physical Review A
- Apr 1, 1991

- arXiv: Optimization and Control
- Dec 12, 2014

- Mathematical Proceedings of the Cambridge Philosophical Society
- Jan 1, 1979

- Annals of Physics
- Nov 1, 2015

- arXiv: Operator Algebras
- Apr 15, 2014

### Quantum Studies: Mathematics and Foundations

- Quantum Studies: Mathematics and Foundations
- Sep 28, 2023

- Quantum Studies: Mathematics and Foundations
- Aug 28, 2023

- Quantum Studies: Mathematics and Foundations
- Aug 19, 2023

- Quantum Studies: Mathematics and Foundations
- Aug 18, 2023

- Quantum Studies: Mathematics and Foundations
- Jun 23, 2023

- Quantum Studies: Mathematics and Foundations
- Jun 12, 2023

- Quantum Studies: Mathematics and Foundations
- May 18, 2023

- Quantum Studies: Mathematics and Foundations
- May 2, 2023

- Quantum Studies: Mathematics and Foundations
- Apr 24, 2023

- Quantum Studies: Mathematics and Foundations
- Apr 24, 2023