Convex Combinations of Pauli Semigroups: Geometry, Measure and an Application

TL;DR

This paper explores the geometry and measure of Markovian and non-Markovian Pauli channels, revealing their non-convexity and quantifying the prevalence of non-Markovian channels within the Pauli simplex.

Contribution

It provides a full characterization of the Pauli simplex, demonstrating the non-convexity of Markovian and non-Markovian channels and quantifying the measure of non-Markovian channels.

Findings

Non-Markovian channels constitute about 87% of the Pauli simplex.

All channels in the Pauli simplex are P divisible.

Neither the set of non-Markovian nor Markovian channels is convex.

Abstract

Finite-time Markovian channels, unlike their infinitesimal counterparts, do not form a convex set. As a particular instance of this observation, we consider the problem of mixing the three Pauli channels, conservatively assumed to be quantum dynamical semigroups, and fully characterize the resulting ``Pauli simplex.'' We show that neither the set of non-Markovian (completely positive indivisible) nor Markovian channels is convex in the Pauli simplex, and that the measure of non-Markovian channels is about 0.87. All channels in the Pauli simplex are P divisible. A potential application in the context of quantum resource theory is also discussed.