Quasi Inverse of Qubit Channels for Mixed States

TL;DR

This paper introduces a method to find a quasi inverse of qubit channels as a unitary map, optimized to minimize trace distance, and extends it to mixed states using a new MSTD-based definition, with results consistent for pure states.

Contribution

It proposes a novel MSTD-based definition of the quasi inverse for mixed states and demonstrates its effectiveness across various qubit channels.

Findings

Quasi inverse as a unitary map minimizes trace distance.

MSTD-based quasi inverse aligns with fidelity-based results for pure states.

Method applicable to multiple qubit channels including Pauli and amplitude damping.

Abstract

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.