Geometry of symmetric and non-invertible Pauli channels

TL;DR

This paper explores the geometric structure of certain Pauli channels, analyzing their properties and classifications using the Hilbert-Schmidt metric, and determining the volume and shape of various subclasses within the space of all Pauli channels.

Contribution

It provides a detailed geometric analysis of symmetric and non-invertible Pauli channels, including volume calculations and shape characterizations of positivity regions.

Findings

Computed relative volumes of entanglement breaking and divisible channels

Determined shapes of complete positivity regions within the Pauli tetrahedron

Analyzed geometry of symmetric and non-invertible Pauli channels

Abstract

We analyze the geometry of positive and completely positive, trace preserving Pauli maps that are fully determined by up to two distinct parameters. This includes five classes of symmetric and non-invertible Pauli channels. Using the Hilbert-Schmidt metric in the space of the Choi-Jamio\lkowski states, we compute the relative volumes of entanglement breaking, time-local generated, and divisible channels. Finally, we find the shapes of the complete positivity regions in relation to the tetrahedron of all Pauli channels.