How to calculate fidelity of a specific quantum channel?

Let $\gamma$ be a completely trace preserving operator such that $\gamma(\rho) \to (1-\epsilon)\rho+\epsilon(|\phi\rangle \langle\phi|)$. Here $\rho$ is density matrix of two dimensional hilbert space and $|\phi\rangle$ a fixed pure state of 2 dimensional hilbert space. The average fidelity of the channel described by some trace preserving operator $\eta$ is given as $F(\eta)=\int \langle\psi| {\eta(|\psi\rangle)} | \psi\rangle d\psi$.

My doubts are

1. Why is average fidelity defined such that we are only taking into account action of $\eta$ on density operator of pure states ( the part $\eta(|\psi\rangle)$ in the formula ). Is it because it would be enough to take average only how pure states are mapped as mixed states are just convex combination of density operator of pure states ?
2. If I am not missing anything and understanding the formula correct then for $\gamma$ $F(\gamma) = (1-\epsilon)+ \epsilon\int\langle \psi|\phi\rangle \langle\phi|\psi\rangle d\psi$, but how do I calculate the second term ?
• I don't want to revamp this from the dead but for future generations yes the formula in point 2 is correct and the "second term" equals $1/d$ where $d$ is the Hilbert's space dimension. This latter term is also called (for self-explanatory reasons) "fidelity of a random guess".
– lcv
Commented Jun 5, 2017 at 22:49