Many-Body Open Quantum Systems

1. An extensive and comprehensive review of a large subset of the field of open quantum many-body systems.
2. It covers the most relevant tools and canonical models, the essential topics (from quantum phase transitions, criticality and symmetry breaking, to dynamics and connections to other fields).
3. It is self contain and an excellent resource for onboarding newcomers (both students and early career scientists, or people from other fields with a knowledge of advanced QM).
4. It is very up to date and written in a clear and pedagogical way.

1. Some areas, such as numerical methods (tensor networks, DMFT, stochastic equations) are described quite succinctly. These may be worse starting points for a newcomer to enter the field.

The authors have developed a large, but very well organized and written lecture notes on the topic of many-body open quantum systems. They focus on a subset of the field that encompasses not only the most fundamental results and ideas, but also the tools, formalisms and canonical models that may help readers enter other areas of research. The presentation is rather exhaustive and covers a broad set of tools, sometimes superficially (e.g. numerical methods), but in general with enough key references to access all methods and concepts.

Overall I find this an extremely useful resource that not only helps the community, but that stands on its own as a wonderful text to quickly onboard beginners (i.e. students and newcomers), allowing them to critically examine topics and tools that might be more interesting for their research.

I strongly recommend publication, with the optional consideration of some scientific and presentation issues I outline below.

# Physics comments:

On page 6, Ref. 41 does not seem to be the expected one. Perhaps the authors refer to J. Resnick, J. Garland, J. Boyd, S. Shoemaker, R. Newrock, Phys. Rev. Lett. 47 (1981) 1542 and P. Martinoli, P. Lerch, C. Leemann, H. Beck, J. Appl. Phys. 26 (1987) 1999.

In Sect. 2.2, the treatment of two-body losses in ultracold atoms (and ultracold molecules) and the emergence of strongly correlated phases is unfair to previous art. Ref. 84 is only discussed as a work engineering losses and not the physics that emerges from it: Zeno suppression of losses and emergence of a 1D TG gas.

A similar problem regards the theory, which ignores early works describing strong two-body losses as as originators of strong correlations in 1D via Zeno physics
https://journals.aps.org/pra/abstract/10.1103/PhysRevA.79.023614
https://iopscience.iop.org/article/10.1088/1367-2630/11/1/013053/meta
The theory in those works is applicable to molecules or atoms and, as a purely Linbladian treatment, predates the non-Hermitian papers cited in Sect. 7.2 These references would also be suitable after Eq. (36), where the dissipative Bose-Hubbard model is introduced without citations and without references to experiments where it has been realized.

In Sect. 2.3, I miss references to a canonical problem in cavities with many atoms, which is the generation of squeezing. This includes experiments without and with feedback, such as https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.104.073602 and https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.116.093602 and is a much simpler model that the, arguably extremely interesting, setups of Bose-Hubbard models in cavities.

On page 67, when discussing tensor networks and star topologies, I feel that there is an unnecessary link to 2020's works, when there were very exhaustive studies of renormalization strategies by Plenio and Huelga, and the development of the MPS with chain mapping algorithms that are of slight more interest. Naturally, such renormalizations have been discovered multiple times and can also be in the DMRG world (the link between TN's and DMRG seems also to be missing, even in passing), as in A. Feiguin's work on spins in electron baths in arbitrary dimensions.

# Style and presentation comments:

- The overall writing is of high quality, but there are sections where the choice of words is somewhat repetitive: e.g., at the end of page 8, the word "discuss(ion)" is used 5 times in 10 lines. Everywhere "however" is also abused and at the end of page 10 the word capture is used three times, the third one with a missing "d" at the end.

- "tracing out the environmental degrees of freedom. [38]." An extra dot there.

- On page 13, instead of Ref. 59 I would have expected the canonical RMP on *single* trapped ions and covers everything, including spontaneous emission, cooling and pumping https://journals.aps.org/rmp/pdf/10.1103/RevModPhys.75.281?casa_token=GnefutWXNywAAAAA%3AXnjSTOTlkxznRNbQLu-yX94QqHS0MMvmH6tfUmPjw3HHXeaVyiLNeQJ0mFMs0Q3cI0yKXx6lZ0T2MQ

- At the end of page 13 and beginning of page 14, the topic of measurement-induced phase transitions is mentioned in a way that seems to make it specific to trapped ions. The authors could add a footnote that this is not specific to ions, though they are particularly well suited, due to quality and speed of measurements.

- Because of the way it is formulated, using a different label for the stochastic variable, the paragraph around (60) seems to apply only to the second unravelling, when it is true in general: the quantum state is the mixed state that results from averaging over realizations. This paragraph is also somewhat redundant with what follows immediately afterwards, unless I am losing some subtletly.

- On page 70, the presentation above Eq. (121) is somewhat confusing. As a student, the subindex "c" is not explained and I would be puzzled by the fact that I have been told the three operator expectation value is zero, but then I am presented with an expansion of that product. I am sure this can be fixed with one or two sentences (which are relevant, since Kubo's cited paper is behind a paywall).

- On page 98, the text "the sum over alpha" probably means "the sum over mu".

- On page 98, this discussion is also ambiguous. In Eq. 167, the authors assume that the eigenvectors will appear in a precise order, when the opposite phase, \rho^{>}-\rho^{<}, is essentially the same vector and eigenstate with the same eigenvalue, but different decomposition 168. Maybe this disambiguation can be referenced or anticipated.

- Fig. 31 does not make it clear what is being plotted. At most, one can deduce a boundary computed by some means, but what this boundary is, is not clear from the figure itself.

Publish (surpasses expectations and criteria for this Journal; among top 10%)