Finite Classical and Quantum Effect Algebras

TL;DR

This paper investigates finite effect algebras, defining classical and quantum types, exploring matrix representations, sum tables, states, and composites, revealing conditions for classicality and the role of atoms in these structures.

Contribution

It introduces new characterizations of classical effect algebras via matrix representations and sum tables, and analyzes their properties and composites in finite cases.

Findings

Classical effect algebras are characterized by a single-row matrix representation.

A sum table provides immediate information about effect sums.

Classical effect algebras are closed under composition.

Abstract

In this article, we only consider finite effect algebras. We define the concepts of classical and quantum effect algebras and show that an effect algebra $E$ is classical if and only if there exists an observable that measures every effect of $E$. We next consider matrix representations of effect algebras and prove an effect algebra is classical if and only if its matrix representation has precisely one row. We then discuss sum table for effect algebras. Although these are not as concise as matrix representations, they give more immediate information about effect sums which are the basic operations of an effect algebra. We subsequently study states on effect algebras and prove that classical effect algebras are quantum effect algebras. Finally, we consider composites of effect algebras. This allows us to study interacting systems described by effect algebras. We show that two effect algebras are classical if and only if their composite is classical. We point out that scale effect algebras are not the only classical effect algebras and stress the importance of atoms in this work.