Is there a no-cloning theorem for open quantum dynamics?

Question

There are numerous papers that prove a no-cloning theorem (or more generally a no-broadcasting theorem) at various levels of generality. However, it is unclear to me if

1. Cloning is proven to be impossible using arbitrary open quantum dynamics (including non-linear dynamics).
2. Broadcasting is proven to be impossible using arbitrary open quantum dynamics (including non-linear dynamics).

Below I define both cloning and broadcasting. Consider a Hilbert space $\mathcal{H} = \mathcal{H}_A \otimes \mathcal{H}_B$ and its associated space of density operators $\mathcal{B}^+(\mathcal{H})$.

Definition (Clone): A state $\rho \in \mathcal{B}^+(\mathcal{H})$ is ideally cloned if there exists a physical process $P$ such that $$\rho \otimes \Sigma \xrightarrow{P} \rho \otimes \rho$$ where $\Sigma$ is some reference state being used to copy $\rho$.

Definition (Broadcast): A state $\rho \in \mathcal{B}^+(\mathcal{H})$ is ideally broadcasted if there exists a physical process $P$ such that $$\rho \otimes \Sigma \xrightarrow{P} \rho^\text{out}$$ such that $$\text{tr}_{A} \rho^{out} = \rho \quad \wedge \quad \text{tr}_{B} \rho^{out} = \rho.$$

Moreover, I provide an example of non-linear quantum dynamics in a usual circumstance. The Gross–Pitaevskii equation.

Answer 1

A channel $\Phi$ allowing to "clone" any input state would mean $\Phi(\rho)=\rho\otimes\rho$ for all states $\rho$.

Any channel $\Phi$ sending states in $\mathbb{C}^n$ to states in $\mathbb{C}^m$ can be represented via an isometry $V:\mathbb{C}^n\to\mathbb{C}^m\otimes\mathbb{C}^{d}$ as $\Phi(X)=\operatorname{tr}_2[VXV^\dagger]$ (this is the so-called Stinespring's dilation).

Thus if $\Phi$ could clone states, you'd have an isometry such that, for all pure states $|\psi\rangle$, $$V|\psi\rangle=|\psi\rangle\otimes|\psi\rangle\otimes|\text{other stuff}\rangle.$$ Note also that any isometry corresponds to a unitary $U$ and some ancillary state $|0\rangle$ via $V=U(I\otimes |0\rangle)$.

In other words, if a channel $\Phi$ can clone states, then there's an isometry/unitary operation that can clone states. Thus you have no-cloning for arbitrary dynamics