# Confusion on the definition of the phase-damping channel

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.

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.

• Well that's a very clever answer. However, I don't see why to define the same process with two different timescales. Could it be said then that Nielsen and Chuang's is the basic definition because it is the simpler one? – Josu Etxezarreta Martinez Nov 27 '18 at 8:39
• @JosuEtxezarretaMartinez: In this case, these maps are standing in for a continuous-time evolution, and in particular describing the evolution after some duration. While $\smash{\mathcal N_{\mathrm P}} = \smash{\mathcal N_{\mathrm{NC}}^2}$, this does not mean that $\smash{\mathcal N_{\mathrm{NC}}^2}$ is in a meaningful way more 'fundamental'. It's less a question of 'defining the same process with two different timescales', and more a question of considering which description of the same process is more helpful for the analysis you want to do (or the presentation of that analysis). – Niel de Beaudrap Nov 27 '18 at 10:38
• Also: on the question of time-scales, consider what map $\smash{\mathcal N_{\mathrm{NC}}}$ realises if you replace $\lambda$ with $\lambda' = 2\lambda - \lambda^2$. Because the difference between Preskill's and Nielsen & Chuang's presentation is just one of time-scale, it follows that there is a re-parameterisation of Nielsen & Chuang's map which realises the same map as Preskill's. The important thing is that these channels are parameterised, and that in principle the parameters would have to be obtained with reference both to a physical model and with some question of how you want to use it. – Niel de Beaudrap Nov 27 '18 at 10:45
• Ok, thanks a lot! That helped to clarify. – Josu Etxezarreta Martinez Nov 27 '18 at 10:46