Convex approximations of quantum channels

TL;DR

This paper develops methods to optimally approximate a quantum channel using convex combinations of available channels, focusing on single-qubit covariant and Pauli channels, with applications to unitary and damping channels.

Contribution

It introduces a convex optimization framework for approximating quantum channels and analyzes specific cases for single-qubit channels, including covariant and Pauli channels.

Findings

Optimal convex mixtures minimize distinguishability from the target channel.

Explicit solutions for approximations of unitary transformations and damping channels.

Framework applicable to various quantum channel approximation problems.

Abstract

We address the problem of optimally approximating the action of a desired and unavailable quantum channel $\Phi $ having at our disposal a single use of a given set of other channels $\{\Psi_i \}$. The problem is recast to look for the least distinguishable channel from $\Phi $ among the convex set $\sum_i p_i \Psi_i$, and the corresponding optimal weights $\{ p_i \}$ provide the optimal convex mixing of the available channels $\{\Psi_i \}$. For single-qubit channels we study specifically the cases where the available convex set corresponds to covariant channels or to Pauli channels, and the desired target map is an arbitrary unitary transformation or a generalized damping channel.