# On optimal cloning and incompatibility

## TL;DR

This paper introduces a new measure of quantum observables' incompatibility based on symmetric quantum cloning machines, establishing bounds and optimal cloning strategies, thus linking no-cloning and incompatibility concepts.

## Contribution

It proposes a cloning-based incompatibility measure, proves its properties, and identifies conditions for optimal cloning of qubit observables, connecting fundamental quantum principles.

## Key findings

- The measure $ ext{Q}_c$ vanishes only for commuting observables.
- An upper bound for $ ext{Q}_c$ is established and attained by mutually unbiased observables.
- The optimal quantum cloner for a pair of qubit observables is derived.

## Abstract

We investigate the role of symmetric quantum cloning machines (QCMs) in quantifying the mutual incompatibility of quantum observables. Specifically, we identify a cloning-based incompatibility measure whereby the incompatibility of a set of observables maybe quantified in terms of how well a uniform ensemble of their eigenstates can be cloned via a symmetric QCM. We show that this new incompatibility measure $\mathcal{Q}_{c}$ is {\it faithful} since it vanishes only for commuting observables. We prove an upper bound for $\mathcal{Q}_{c}$ for any set of observables in a finite-dimensional system and show that the upper bound is attained if and only if the observables are mutually unbiased. Finally, we use our formalism to obtain the optimal quantum cloner for a pair of qubit observables. Our work marks an important step in formalising the connection between two fundamental concepts in quantum information theory, namely, the no-cloning principle and the existence of incompatible observables in quantum theory.