Information-theoretic limitations on approximate quantum cloning and broadcasting

TL;DR

This paper establishes new quantitative limits on approximate quantum cloning and broadcasting of mixed states using information-theoretic methods, revealing dualities between cloning machines and partial trace channels, and exploring implications for quantum information recovery.

Contribution

It generalizes no-cloning and no-broadcasting theorems by providing quantitative bounds and introduces a duality framework between cloning and partial trace channels, extending to non-symmetric subspaces.

Findings

Derived new bounds on approximate cloning of mixed states.

Revealed duality between universal cloning machines and partial trace channels.

Demonstrated control over cloning performance using a-priori information.

Abstract

We prove new quantitative limitations on any approximate simultaneous cloning or broadcasting of mixed states. The results are based on information-theoretic (entropic) considerations and generalize the well known no-cloning and no-broadcasting theorems. We also observe and exploit the fact that the universal cloning machine on the symmetric subspace of $n$ qudits and symmetrized partial trace channels are dual to each other. This duality manifests itself both in the algebraic sense of adjointness of quantum channels and in the operational sense that a universal cloning machine can be used as an approximate recovery channel for a symmetrized partial trace channel and vice versa. The duality extends to give control on the performance of generalized UQCMs on subspaces more general than the symmetric subspace. This gives a way to quantify the usefulness of a-priori information in the context of cloning. For example, we can control the performance of an antisymmetric analogue of the UQCM in recovering from the loss of $n-k$ fermionic particles.