# Positivity and complete positivity of differentiable quantum processes

## TL;DR

This paper investigates the conditions under which quantum processes, modeled as differentiable CPTP maps, are divisible into CP or P types, linking these properties to quantum non-Markovianity and providing geometric criteria for specific classes.

## Contribution

It introduces new conditions for CP- and P-divisibility of quantum processes using generator representations and Sylvester criterion, with explicit results for single qubit and certain channel classes.

## Key findings

- CP- and P-divisibility depend on the dissipation matrix in the master equation.
- Derived geometric inequalities for non-unital anisotropic Pauli channels.
- Established connections between divisibility and quantum non-Markovianity.

## Abstract

We study quantum processes, as one parameter families of differentiable completely positive and trace preserving (CPTP) maps. Using different representations of the generator, and the Sylvester criterion for positive semi-definite matrices, we obtain conditions for the divisibility of the process into completely positive (CP-divisibility) and positive (P-divisibility) infinitesimal maps. Both concepts are directly related to the definition of quantum non-Markovianity. For the single qubit case we show that CP- and P-divisibility only depend on the dissipation matrix in the master equation form of the generator. We then discuss three classes of processes where the criteria for the different types of divisibility result in simple geometric inequalities, among these the class of non-unital anisotropic Pauli channels.