Quasi-inversion of qubit channels

In general quantum operations, or quantum channels cannot be inverted by physical operations, i.e., by completely positive trace-preserving maps. An arbitrary state passing through a quantum channel loses its fidelity with the input. Given a quantum channel epsilon, we discuss the concept of its quasi-inverse as a completely positive trace-preserving map epsilon(qi) which when composed with epsilon increases its average input-output fidelity in an optimal way. The channel epsilon(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.