Observables on Quantum Structures

TL;DR

This paper investigates how partial information about observables on various quantum structures, specifically for intervals of the form (-infinity, t), can uniquely determine the entire observable, extending classical measure theory concepts.

Contribution

It demonstrates that partial data on certain intervals suffices to uniquely identify observables across multiple quantum and effect algebra structures.

Findings

Partial information on intervals determines the entire observable.

Unique determination of observables on diverse quantum structures.

Applicable to effect algebras, Hilbert space operators, and effect-tribes.

Abstract

An observable on a quantum structure is any $\sigma$-homomorphism of quantum structures from the Borel $\sigma$-algebra into the quantum structure. We show that our partial information on an observable known only for all intervals of the form $(-\infty,t)$ is sufficient to determine uniquely the whole observable defined on quantum structures like $\sigma$-MV-algebras, $\sigma$-effect algebras, Boolean $\sigma$-algebras, monotone $\sigma$-complete effect algebras with the Riesz Decomposition Property, the effect algebra of effect operators of a Hilbert space, and a system of functions, and an effect-tribe.