Olson Order of Quantum Observables

TL;DR

This paper introduces the Olson order on bounded quantum observables within a complete lattice effect algebra, demonstrating that this set forms a Dedekind complete lattice, extending Olson's spectral order concept.

Contribution

It defines a new partial order called Olson order on bounded observables and proves the set forms a Dedekind complete lattice, advancing the mathematical structure of quantum observables.

Findings

The set of bounded observables forms a Dedekind complete lattice.

The Olson order extends spectral order to bounded observables.

Provides a new framework for ordering quantum effects.

Abstract

Using ideas of Olson \cite{Ols} who showed that the system of effect operators of a Hilbert space can be ordered by the so-called spectral order such that the system of effect operators is a complete lattice. Using his ideas, we introduce a partial order, called the Olson order, on the set of bounded observables of a complete lattice effect algebra. We show that the set of bounded observables is a Dedekind complete lattice.0