What's the reasoning behind writing the isometric representation of a channel?

Question

I am reading about phase damping channel from Preskill's notes. He writes off the unitary representation of the channel as

Unitary representation. An isometric representation of the channel is \begin{equation} |0\rangle_{A} \rightarrow \sqrt{1-p}|0\rangle_{A}\otimes|0\rangle_{E}+\sqrt{p}|0\rangle_{A}\otimes|1\rangle_{E} \end{equation} \begin{equation} |1\rangle_{A} \rightarrow \sqrt{1-p}|1\rangle_{A}\otimes|0\rangle_{E}+\sqrt{p}|1\rangle_{A}\otimes|2\rangle_{E} \end{equation} In this case, unlike the depolarizing channel, qubit $A$ does not make any transitions in the $\left\{|0\rangle,|1\rangle\right\}$ basis. Instead, the environment "scatters" of the qubit occasionally (with probability $p$), being kicked into the state $|1\rangle_E$ if $A$ is in the state $|0\rangle_A$ and into the state $|2\rangle_E$ if $A$ is in the state $|1\rangle_A$. Furthermore, also unlike the depolarizing channel, the channel picks out a preferred basis for qubit $A$; the basis $\left\{|0\rangle,|1\rangle\right\}$ is the only basis in which bit flips never occur.

However, I am unable to understand the reasoning behind writing off the isometric representation in this manner. Preskill does not explain why the isometric representation should be in this way. Also he claims that the channel "picks out a preferred basis for qubit $A$" which too is not evident to me. It would be helpful if someone explains the reasoning behind these.

Answer 1

This is an example of a Steinspring dilation. The idea is that any channel $\Phi$ can be represented as $\Phi(\rho)=\operatorname{tr}_2[V\rho V^\dagger]$ for some isometry $V$. You can imagine $V$ as being the matrix obtained stacking the Kraus operators of the channel one on top of the other, see e.g. this post.

More precisely, if you have a Kraus decomposition $\Phi(\rho)=\sum_k A_k \rho A_k^\dagger$, then $$V|\psi\rangle\equiv \sum_k (A_k |\psi\rangle)\otimes |k\rangle\tag1$$ works as a dilation isometry. What the book means writing those equations is to represent the channel $\Phi$ as the action of the isometry $V$. They are literally writing (1) in the special case of the phase damping channel, with Kraus operators $$A_0 = \sqrt{1-p} I, \quad A_1 = \sqrt p |0\rangle\!\langle 0|, \quad A_2=\sqrt p |1\rangle\!\langle 1|.$$ You might in fact find a more efficient (as in, requiring a smaller ancillary space) from this: the phase damping channel can be represented using only two Kraus operators: $B_1=\sqrt{2-p}I$ and $B_2=\sqrt p Z$, which correspond to the isometric representation $$|\psi\rangle\to\sqrt{2-p}|\psi\rangle|0\rangle+\sqrt p Z|\psi\rangle|1\rangle,$$ or equivalently, $$|0\rangle\to\sqrt{2-p} |0\rangle|0\rangle + \sqrt p |0\rangle|1\rangle, \\ |1\rangle\to\sqrt{2-p} |1\rangle|0\rangle - \sqrt p |1\rangle|1\rangle.$$ You can directly verify that these result in an identical evolution on the system.