# Measure of positive and not completely positive single-qubit Pauli maps

## TL;DR

This paper investigates the relative measure of positive and not-completely positive (NCP) single-qubit Pauli maps, revealing that NCP maps occupy twice the volume of CP maps within a specific equivalence class space.

## Contribution

It introduces a method to compare the volumes of CP and NCP maps for single-qubit Pauli channels, showing NCP maps are twice as prevalent as CP maps in the considered space.

## Key findings

- NCP maps have twice the volume of CP maps in the studied space.
- The analysis is restricted to an equivalence class of Pauli channels.
- The work provides insight into the prevalence of NCP maps in quantum dynamics.

## Abstract

The time evolution of an initially uncorrelated system is governed by a completely positive (CP) map. More generally, the system may contain initial (quantum) correlations with an environment, in which case the system evolves according to a not-completely positive (NCP) map. It is an interesting question what the relative measure is for these two types of maps within the set of positive maps. After indicating the scope of the full problem of computing the true volume for generic maps acting on a qubit, we study the case of Pauli channels in an abstract space whose elements represent an equivalence class of maps that are identical up to a non-Pauli unitary. In this space, we show that the volume of NCP maps is twice that of CP maps.