The Geometry of Single-Qubit Maps

TL;DR

This paper explores the geometric structure of single-qubit quantum channels, providing a canonical form for unital channels and insights into their approximation and generation methods.

Contribution

It introduces a canonical form for all unital single-qubit channels and links their geometry to approximation and dynamical generation techniques.

Findings

Canonical form for unital single-qubit channels

Geometric understanding of positive map approximation

Connection to dynamical generation of quantum channels

Abstract

The physically allowed quantum evolutions on a single qubit can be described in terms of their geometry. From a simple parameterisation of unital single-qubit channels, the canonical form of all such channels can be given. The related geometry can be used to understand how to approximate positive maps by completely-positive maps, such as in the case of optimal eavesdropping strategies. These quantum channels can be generated by the appropriate network or through dynamical means. The Str{\o}mer-Woronowisc result can also be understood in terms of this geometry.