Geometry of depolarizing channels

TL;DR

This paper investigates the geometric structure of depolarizing channels in quantum systems, characterizing their compression domains for three and four level systems, and conjecturing a general pattern for higher dimensions.

Contribution

It provides a detailed geometric analysis of depolarizing channels for specific quantum systems and proposes a conjecture for the structure in higher-dimensional systems.

Findings

Depolarizing maps form a convex region in the space of compression coefficients.

The region is a curved surface for three-level systems.

The region is a simplex for four-level systems, with a conjecture for higher levels.

Abstract

Depolarizing maps acting on an N dimensional system are completely positive maps resulting into compression of the Bloch ball along the different polarization directions. In the qubit case these maps are a convex sum of four extremal maps and form a simplex in the space of compression coefficients along the three polarization directions. We calculate the compression domain for three and four level systems. For a three level system the region has curved surfaces, but it is a simplex for a four level system. We conjecture that it is a simplex in the case of 2^n level systems.