Imperfect cloning operations in algebraic quantum theory

TL;DR

This paper investigates the possibility of imperfect cloning in algebraic quantum theory, establishing that universal $ ext{ extepsilon}$-imperfect cloning operations are impossible unless the system's algebra is Abelian, which it is not in typical quantum cases.

Contribution

It introduces a formal definition of $ ext{ extepsilon}$-imperfect cloning in a C*-algebraic framework and proves its non-existence in non-Abelian systems when fidelity loss is below 1/4.

Findings

Universal $ ext{ extepsilon}$-imperfect cloning is impossible in non-Abelian quantum systems.

Cloning is only possible if the algebra of observables is Abelian.

The result extends the no-cloning theorem to approximate cloning in algebraic quantum theory.

Abstract

No-cloning theorem says that there is no unitary operation that makes perfect clones of non-orthogonal quantum states. The objective of the present paper is to examine whether an imperfect cloning operation exists or not in a C*-algebraic framework. We define a universal $\epsilon$-imperfect cloning operation which tolerates a finite loss $\epsilon$ of fidelity in the cloned state, and show that an individual system's algebra of observables is Abelian if and only if there is a universal $\epsilon$-imperfect cloning operation in the case where the loss of fidelity is less than 1/4. Therefore, in this case no universal $\epsilon$-imperfect cloning operation is possible in algebraic quantum theory.