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 $\sigma$?


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