Consider a $2$-qubit system with Hilbert space $\mathcal{H} \cong \mathcal{H}_1 \otimes \mathcal{H}_2$. A pure separable state in $\mathcal{H}$ is of the form $\lvert \psi_1 \rangle \otimes \lvert \psi_2 \rangle$ where $\lvert \psi_i \rangle \in \mathcal{H}_i$. The associated density matrix is $\rho \equiv \lvert \psi_1 \rangle \langle \psi_1 \lvert \otimes \lvert \psi_2 \rangle \langle \psi_2 \lvert$.
Consider a unitary operator $U: \mathcal{H} \to \mathcal{H}$ that is not of the form $U_1 \otimes U_2$ where $U_i: \mathcal{H}_i \to \mathcal{H}_i$.
Does there exist such a $U$ such that $U\rho U^\dagger = \lvert \psi'_1 \rangle\langle \psi'_1 \lvert \otimes \lvert \psi'_2 \rangle \langle \psi'_2\lvert \equiv \rho'$ where $\lvert \psi'_i \rangle \in \mathcal{H}_i$ and $\rho \neq \rho'$? In words, does there exist a non-local unitary operator that maps a particular pure product density matrix $\rho$ to a distinct pure product density matrix $\rho'$?
I throw out the case in which $\rho = \rho'$ above because it is conceivable that if $U$ stabilizes $\rho$ $$U\rho U^\dagger = \rho,$$ then $U$ could be non-local.
