# Is a quantum channel reversible if all Kraus operators are proportional to unitaries?

In preskill's online lecture p.13, he stated that if a channel is reversible, i.e., $$\varepsilon^{-1}\circ\varepsilon(\rho)=\rho$$ for any $$\rho$$, then the kraus operator of the quantum channel must be proportional to unitary operator. I know the reverse might not be true, but take dephasing channel as an example, the kraus operators of the dephasing channel are: $$E_1=\sqrt{p}I \\ E_2=\sqrt{1-p}Z,$$ where $$Z$$ stands for pauli operator. So, does this quantum channel reversible, since both $$I$$ and $$Z$$ are unitary?

• But why in this paper, the author stated that the similar process using $\varepsilon(s,0)^{-1}$(Eq. 6), obviously showing the idea of the inverse of the map? Aug 17 at 12:58