# A unitary that induces Entanglement Breaking Channel for any ancillary state

Let's say there is a unitary operator $$M:\mathcal{H}_1\otimes\mathcal{H}_2\to\mathcal{H}_A\otimes\mathcal{H}_B$$ such that the channel $$\Phi$$ given as $$\Phi(\rho)=\text{Tr}_BM(\rho\otimes\phi)M^\dagger$$ is always an Entanglement-Breaking (EB) channel for any pure state $$\phi$$ on $$\mathcal{H}_2$$. Does this condition give any constraints to the structure of the unitary operator $$M$$?

As a question in the similar vein, let's say there is a channel given in the form $$\Psi(\rho)=V(\rho\otimes\sigma)V^\dagger$$ for a fixed mixed state $$\sigma$$, and this channel $$\Psi$$ always outputs separable state. Does this give any nontrivial constraint to $$V$$ or $$\sigma$$?