# Phase spaces that cannot be cloned in classical mechanics

## TL;DR

This paper explores the concept of cloning in classical mechanics using symplectic geometry, establishing conditions under which phase spaces can or cannot be cloned, and connecting these results to classical measurement issues.

## Contribution

It introduces a natural definition of cloning in classical mechanics, proves that only contractible phase spaces can be cloned, and extends the concept to approximate cloning.

## Key findings

- Phase spaces like N can be cloned exactly.
- Non-contractible phase spaces cannot be cloned.
- Approximate cloning is also restricted to contractible spaces.

## Abstract

The quantum no cloning theorem is an essential result in quantum information theory. Following this idea, we give a physically natural definition of cloning in the context of classical mechanics using symplectic geometry, building on work of Fenyes. We observe, following Fenyes, any system with phase space $(\mathbb{R}^{2N}, dx_i\wedge dy_i)$ can be cloned in our definition. However, we show that if $(M,\omega)$ can be cloned in our definition, then $M$ must be contractible. For instance, this shows the simple pendulum cannot be cloned in Hamiltonian mechanics. We further formulate a robust notion of approximate cloning, and show that if $(M, \omega)$ can be approximately cloned, then $M$ is contractible. We give interpretations of our results and in some special cases reconcile our no cloning theorems with the general experience that classical information is clonable. Finally we point to new directions of research, including a connection of our result with the classical measurement problem.