Finite (quantum) effect algebras

TL;DR

This paper classifies finite effect algebras, characterizes their sums, and identifies quantum effect algebras among small cases, revealing structural and quantum properties.

Contribution

It provides a complete classification of finite effect algebras, characterizes scale effect algebras, and identifies quantum effect algebras with finite states.

Findings

Finite effect algebra sums range from n-2 to (n-1)(n-2)/2.

Scale effect algebras are characterized as those with maximal sums.

Quantum effect algebras are identified by finite order-determining states.

Abstract

We investigate finite effect algebras and their classification. We show that an effect algebra with $n$ elements has at least $n-2$ and at most $(n-1)(n-2)/2$ nontrivial defined sums. We characterize finite effect algebras with these minimal and maximal number of defined sums. The latter effect algebras are scale effect algebras (i.e., subalgebras of [0,1]), and only those. We prove that there is exactly one scale effect algebra with $n$ elements for every integer $n \geq 2$. We show that a finite effect algebra is quantum effect algebra (i.e. a subeffect algebra of the standard quantum effect algebra) if and only if it has a finite set of order-determining states. Among effect algebras with 2-6 elements, we identify all quantum effect algebras.