Geometry of the Pauli maps and Pauli channels

TL;DR

This paper explores the geometric structure of trace-preserving Pauli maps and channels, providing analytical volume calculations and insights into their entanglement-breaking, divisibility, and Markovian properties.

Contribution

It offers a detailed geometric analysis of Pauli maps using the Choi-Jamiołkowski isomorphism, including volume computations and characterizations of various channel properties.

Findings

Computed volumes of positive, trace-preserving Pauli maps and channels.

Identified the relative volumes of entanglement-breaking Pauli channels.

Analyzed the divisibility and Markovianity properties of Pauli channels.

Abstract

We analyze the geometrical properties of trace-preserving Pauli maps. Using the Choi-Jamio\lkowski isomorphism, we express the Hilbert-Schmidt line and volume elements in terms of the eigenvalues of the Pauli map. We analytically compute the volumes of positive, trace-preserving Pauli maps and Pauli channels. In particular, we find the relative volumes of the entanglement breaking Pauli channels, as well as the channels that can be generated by a time-local generator. Finally, we show what part of the Pauli channels are P and CP-divisible, which is related to the notion of Markovianity.