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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?

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No. The key thing about what Preskill is saying is that all the Kraus operators must be proportional to the same unitary. Your two Kraus operators are proportional to different Kraus operators.

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  • $\begingroup$ 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? $\endgroup$
    – Sherlock
    Aug 17 at 12:58
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    $\begingroup$ I think the resolution is that Preskill is talking about if there exists a physical process that can implement the inverse (and therefore a set of Krauss operators exists for it), while that paper is just requiring the existence of a mathematical entity (matrix) that can invert the operation. $\endgroup$
    – DaftWullie
    Aug 17 at 13:24
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    $\begingroup$ Try visualising depolarising noise on a Bloch sphere. This takes states on the surface of the Bloch sphere, and shrinks them. We know that we cannot physically reverse the process because you cannot reduce the mixedness. However, we mathematically know how to enlarge the shrunken ball to return it to the surface of the sphere. $\endgroup$
    – DaftWullie
    Aug 17 at 13:25

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