No-Cloning in Reduced Power Algebras

TL;DR

This paper demonstrates that the No-Cloning property in quantum mechanics remains valid when reformulated within broader scalar frameworks called reduced power algebras, extending its applicability beyond standard complex numbers.

Contribution

It shows that the No-Cloning theorem holds in quantum frameworks based on reduced power algebras, not just traditional complex numbers, broadening the mathematical foundations of quantum theory.

Findings

No-Cloning property is preserved in reduced power algebra frameworks.

Quantum mechanics can be reformulated with alternative scalar systems.

The validity of No-Cloning extends beyond standard complex number-based quantum mechanics.

Abstract

The No-Cloning property in Quantum Computation is known not to depend on the unitarity of the operators involved, but only on their linearity. Based on that fact, here it is shown that the No-Cloning property remains valid when Quantum Mechanics is re-formulated within far wider frameworks of {\it scalars}, namely, one or the other of the infinitely many {\it reduced power algebras} which can replace the usual real numbers $\mathbb{R}$, or complex numbers $\mathbb{C}$.