Ignorance is a bliss: mathematical structure of many-box models

TL;DR

This paper explores the mathematical structure of many-box models, revealing they are set-representable effect algebras, and discusses their relation to quantum models and the Local Orthogonality principle.

Contribution

It establishes the propositional system of many-box models as set-representable effect algebras and clarifies their non-equivalence to quantum models, challenging the pursuit of principles to recover quantum correlations.

Findings

Many-box models' propositional systems are set-representable effect algebras.

2-box models form orthomodular posets; 1-box models form orthomodular lattices.

Box models cannot be considered as generalizations of quantum models.

Abstract

We show that the propositional system of a many-box model is always a set-representable effect algebra. In particular cases of 2-box and 1-box models it is an orthomodular poset and an orthomodular lattice respectively. We discuss the relation of the obtained results with the so-called Local Orthogonality principle. We argue that non-classical properties of box models are the result of a dual enrichment of the set of states caused by the impoverishment of the set of propositions. On the other hand, quantum mechanical models always have more propositions as well as more states than the classical ones. Consequently, we show that the box models cannot be considered as generalizations of quantum mechanical models and seeking for additional principles that could allow to "recover quantum correlations" in box models is, at least from the fundamental point of view, pointless.