Confusion on the definition of the phase-damping channel

Question

I am reading about the phase damping channel, and I have seen that some of the different references talking about such channel give different definitions of the Kraus operators that define the action of such channel.

For example, Nielsen and Chuang define in page 384 the phase damping channel with Kraus operators \begin{equation} E_0=\begin{pmatrix}1 & 0 \\ 0 & \sqrt{1-\lambda}\end{pmatrix}, \qquad E_1=\begin{pmatrix}0 & 0\\0 & \sqrt{\lambda} \end{pmatrix}, \end{equation} where $\lambda$ is the phase damping parameter. However, in the $28^{th}$ page of Preskill's notes on quantum error correction, such channel is defined by Kraus operators: \begin{equation} E_0=\sqrt{1-\lambda}I, \qquad E_1=\begin{pmatrix}\sqrt{\lambda} & 0 \\0 & 0 \end{pmatrix}, \qquad E_2=\begin{pmatrix} 0 & 0 \\ 0 & \sqrt{\lambda}\end{pmatrix}. \end{equation}

Seeing the notable difference between both descriptions, while also having a diffeent number of Kraus operators, I am wondering which is the correct one, or if they are equivalent, why is it such case. A unitary description of the phase damping channel will also be helpful for me.

Answer 1

Let $\mathcal{N}$ be the channels which subscripts for which conventions.

$$ \mathcal{N}_{N.C.} (\rho) = \begin{pmatrix} \rho_{00} & \rho_{01} \sqrt{1-\lambda}\\ \rho_{10} \sqrt{1-\lambda} & \rho_{11} \end{pmatrix} $$

As compared to

$$ \mathcal{N}_{P} (\rho) = \begin{pmatrix} \rho_{00} & \rho_{01} (1-\lambda)\\ \rho_{10} (1-\lambda) & \rho_{11} \end{pmatrix} $$

So you can see that $\mathcal{N}_P (\bullet) = \mathcal{N}_{N.C.} (\mathcal{N}_{N.C.} (\bullet ))$

Easy to see that these are both representing the same sort of process just with different timescales. Preskill's does Nielsen-Chuang's twice.