Are two outputs of a quantum operation (CPTP map) themselves related by a quantum operation?

Question

$\def\ket#1{|#1\rangle} \def\bra#1{\langle#1|} \def\mt#1{\mathrm{#1}} \def\E{\mathcal E} \def\F{\mathcal F}$ Let $\ket\phi$ and $\ket\psi$ be pure states on the same quantum system, so that $\ket\psi=\mt U\ket\phi$ for some unitary $\mt U$. Say we have a quantum operation (CPTP map) $\E$, and write $\sigma_\phi=\E(\ket\phi\bra\phi)$, $\sigma_\psi=\E(\ket\psi\bra\psi)$.

Question: given that $\ket\psi$ and $\ket\phi$ are related by a unitary $\mt U$, are $\sigma_\psi$ and $\sigma_\phi$ related by a quantum operation? I.e. does there exist a quantum operation $\F$ such that $\sigma_\psi=\F(\sigma_\phi)$ (or the other way around)?

My feeling says this has to be true, but I'm struggling to prove it.

Answer 1

Two states are always related by a CPTP map, regardless where they come from.

Just take e.g. $$\mathcal F(\rho) =\mathrm{tr}(\rho) \, \sigma_\psi\ . $$