Dissipative flow equations

- The method is generally well described and detailed in the main text, and several appendices complete it very well.
- The manuscript presents 4 applications of the method in different contexts.

- The language is sometimes difficult to understand for a physicist more familiar with standard open quantum system methods. The connection could be slightly clearer.
- The method derived in the manuscript does not provide at this point a clear advantage compared to other existing methods, but has potential for future works.

The manuscript presents the generalization of the flow equations to open quantum systems described via standard Markovian Lindblad master equation. After a well-motivated introduction, the authors describe the spirit of flow equations, define and derive the corresponding related quantities and finally apply the methods in four different contexts.

While the method derived and used in the manuscript does not provide an advantage compared to other existing methods to solve the investigated problems, the authors mention possible future interesting perspectives for future works.

I would thus recommend the manuscript for publication in SciPost, as I believe this manuscript would be a solid basis for future applications of the method, provided the authors address the following (small) comments/suggestions.

1. In the presentation of the different generators $\eta$, the authors rely a few times on a perturbative treatment involving some parameter $\xi$. In order to make a clearer connection to concrete physical problems, the authors may want to give a concrete physical interpretation of $\xi$ in the case of one or two simple examples.

2. In the first example of section 4, I find that the explanation of the procedure to obtain the results plotted in Fig. 1-4 could be clearer.
- What could correspond to the non-Hermitian matrix A (a Liouvillian or a non-Hermitian Hamiltonian ?) ?
- What 'flow equations' presented above are concretely solved ? Some references to the equations could be useful.
- the 'flow step' in Section 4 is denoted by $d$ and by $d\ell$ in Section 6.2. The expression $d\ell$ is probably better.

3. The authors focus mainly on the diagonalization of the Liouvillian in the four different examples. I would suggest the authors provide (e.g. in the second example) the time-evolution and/or steady state of the density matrix that correspond to the parametrization (42) of the Liouvillian, in order to provide a slightly more physical picture.

4. While Lindblad master equations to describe open quantum system dynamics are common and relevant in many situations, it is sometimes necessary to go beyond the standard Born-Markov approximations to describe non-Markovian effects (e.g. in the solid-state). Would it be possible to generalize the flow equations to these situations (e.g. non-Markovian master equations (time non-local master equations) or more simply Redfield master equations ( time-local but non-positive master equations (see e.g. [1]) ? The authors may want to provide comments on that, as it could provide potentially more interest in the development of the method.

[1] R. Hartmann and W. T. Strunz, Accuracy assessment of perturbative master equations: Embracing nonpositivity, Phys. Rev. A 101, 012103 (2020).