Convex Combinations of Pauli Semigroups: Geometry, Measure and an Application
Vinayak Jagadish, R. Srikanth, Francesco Petruccione

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.
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