# How to find the fidelity between two state when one is an operator?

I am going through Nielsen and Chuang and am finding the chapter on error-correction particularly confusing. At the moment I am stuck on exercise 10.12 which states

Show that the fidelity between the state $|0 \rangle$ and $\varepsilon(|0\rangle\langle0|)$ is $\sqrt{1-2p\backslash3}$, and use this to argue that the minimum fidelity for the depolarizing channel is $\sqrt{1-2p\backslash3}$.

As I understand $\varepsilon$ is a quantum operation and could be whatever we want as long as it fits the definition, do I assume $\varepsilon$ is the depolarizing channel or is there some general operation I don't know about.

Thanks!

The channel $\mathcal{E}$ is explicitly defined in the preceding paragraph as being the depolarising channel. Thus, all you need to calculate is $$F=\sqrt{\langle 0|\mathcal{E}(|0\rangle\langle 0|)|0\rangle}.$$