Quasi inversion of qubit channels

Quantum operations, or quantum channels cannot be inverted in general. An arbitrary state passing through a quantum channel looses its fidelity with the input. Given a quantum channel ${\cal E}$, we introduce the concept of its quasi-inverse as a map ${\cal E}^{qi}$ which when composed with ${\cal E}$ increases its average input-output fidelity in an optimal way. The channel ${\cal E}^{qi}$ comes as close as possible to the inverse of a quantum channel. We give a complete classification of such maps for qubit channels and provide quite a few illustrative examples.