Quasi Inverse of Qubit Channels for Mixed States

We found the quasi inverse of qubit channels as a unitary map, $\mathcal{E}^i$, by minimizing the average trace distance between the input state to the channel and the output of the quasi inverse channel for arbitrary qubit channel $\mathcal{E}$ and for arbitrary input states. The channel $\mathcal{E}$ was assumed completely positive and trace-preserving. To find the quasi inverse for mixed states, we proposed an alternative definition of the quasi inverse based on the mean square of the trace distance (MSTD) of a channel. The definition based on the trace distance allowed easy generalization of the quasi inverse to mixed input states. The quasi inverse of the Pauli, generalized amplitude damping, mixed unitary, and tetrahedron channels calculated based on the MSTD agreed with the one computed using average fidelity in the special case of input states being pure.