# Adding dynamical generators in quantum master equations

## TL;DR

This paper investigates the conditions under which dynamical generators in quantum master equations can be added without losing physical validity, highlighting limitations and providing criteria for justified addition.

## Contribution

It establishes a general condition for the legitimate addition of generators in quantum master equations, especially under weak coupling and commuting Hamiltonian and interactions.

## Key findings

- Generators can be added without losing physicality only under restrictive conditions.
- Naively adding generators may not reflect the true system evolution in general.
- Counterexamples show the exact dynamics often differ from the generator sum approximation.

## Abstract

The quantum master equation is a widespread approach to describing open quantum system dynamics. In this approach, the effect of the environment on the system evolution is entirely captured by the dynamical generator, providing a compact and versatile description. However, care needs to be taken when several noise processes act simultaneously or the Hamiltonian evolution of the system is modified. Here, we show that generators can be added at the master equation level without compromising physicality only under restrictive conditions. Moreover, even when adding generators results in legitimate dynamics, this does not generally correspond to the true evolution of the system. We establish a general condition under which direct addition of dynamical generators is justified, showing that it is ensured under weak coupling and for settings where the free system Hamiltonian and all system-environment interactions commute. In all other cases, we demonstrate by counterexamples that the exact evolution derived microscopically cannot be guaranteed to coincide with the dynamics naively obtained by adding the generators.