No-Cloning Theorem on Quantum Logics

TL;DR

This paper explores the conditions under which quantum and effect algebras allow cloning operations, showing that only classical (Boolean) structures permit cloning, thus highlighting fundamental differences between classical and quantum logics.

Contribution

It proves that only Boolean algebras admit cloning in a logico-algebraic framework, extending the no-cloning theorem to effect algebras and linking cloning to hidden variables.

Findings

Cloning is only possible in Boolean algebras.

Atomic Archimedean effect algebras admitting cloning are Boolean.

Partial relation between cloning on effect algebras and hidden variables.

Abstract

This paper discusses the no-cloning theorem in a logico-algebraic approach. In this approach, an orthoalgebra is considered as a general structure for propositions in a physical theory. We proved that an orthoalgebra admits cloning operation if and only if it is a Boolean algebra. That is, only classical theory admits the cloning of states. If unsharp propositions are to be included in the theory, then a notion of effect algebra is considered. We proved that an atomic Archimedean effect algebra admitting cloning operation is a Boolean algebra. This paper also presents a partial result indicating a relation between cloning on effect algebras and hidden variables.