Algorithmic No-Cloning Theorem

TL;DR

This paper introduces an algorithmic perspective to quantum information, establishing conservation laws and limitations on cloning quantum states based on their algorithmic information content.

Contribution

It presents an algorithmic version of the no-cloning theorem and explores properties of quantum states' algorithmic information and rarity.

Findings

Most pure states have minimal self-algorithmic information.

Only a small subset of states can clone significant algorithmic information.

Rarity of states does not increase under POVM measurements.

Abstract

We introduce the notions of algorithmic mutual information and rarity of quantum states. These definitions enjoy conservation inequalities over unitary transformations and partial traces. We show that a large majority of pure states have minute self algorithmic information. We provide an algorithmic variant to the no-cloning theorem, by showing that only a small minority of quantum pure states can clone a non negligible amount of algorithmic information. We also provide a chain rule inequality for quantum algorithmic entropy. We show that rarity does not increase under POVM measurements.