Finite Classical and Quantum Effect Algebras

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.