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

$$\def\ket#1{|#1\rangle} \def\bra#1{\langle#1|} \def\mt#1{\mathrm{#1}} \def\E{\mathcal E} \def\F{\mathcal F}$$ 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.

## 1 Answer

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

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

• Ah right, this indeed works. Thanks! Commented Aug 25, 2020 at 13:36
• Blatant self-advertisement: Canonical examples of quantum channels, #5 ;) Commented Aug 25, 2020 at 17:59