We find 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 an arbitrary qubit channel $\mathcal{E}$ and for arbitrary input states. The channel $\mathcal{E}$ is assumed to be completely positive and trace preserving. To find the quasi-inverse for mixed states, we propose an alternative definition of the quasi-inverse based on the mean-square trace distance (MSTD) of a channel. The definition based on the trace distance allows 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 agrees with the one computed using average fidelity in the special case of input states being pure.

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• Quantum information processing
• Quantum information theory