(Modular) Effect Algebras are Equivalent to (Frobenius) Antispecial Algebras

TL;DR

This paper establishes an equivalence between effect algebras and Frobenius antispecial algebras, revealing deep connections between quantum logic and categorical quantum mechanics, and offering new insights into quantum structures.

Contribution

It demonstrates that effect algebras and Frobenius antispecial algebras are mathematically equivalent, linking quantum logic laws with categorical quantum mechanics conditions.

Findings

Effect algebra units are each other's unique complements.

The modularity law corresponds to the Frobenius condition.

Effect algebras and Frobenius antispecial algebras are equivalent.

Abstract

Effect algebras are one of the generalizations of Boolean algebras proposed in the quest for a quantum logic. Frobenius algebras are a tool of categorical quantum mechanics, used to present various families of observables in abstract, often nonstandard frameworks. Both effect algebras and Frobenius algebras capture their respective fragments of quantum mechanics by elegant and succinct axioms; and both come with their conceptual mysteries. A particularly elegant and mysterious constraint, imposed on Frobenius algebras to characterize a class of tripartite entangled states, is the antispecial law. A particularly contentious issue on the quantum logic side is the modularity law, proposed by von Neumann to mitigate the failure of distributivity of quantum logical connectives. We show that, if quantum logic and categorical quantum mechanics are formalized in the same framework, then the antispecial law of categorical quantum mechanics corresponds to the natural requirement of effect algebras that the units are each other's unique complements; and that the modularity law corresponds to the Frobenius condition. These correspondences lead to the equivalence announced in the title. Aligning the two formalisms, at the very least, sheds new light on the concepts that are more clearly displayed on one side than on the other (such as e.g. the orthogonality). Beyond that, it may also open up new approaches to deep and important problems of quantum mechanics (such as the classification of complementary observables).