Reply to Referee 1 We thank the Referee for their time in reviewing our manuscript, as well as for the questions and comments they have made. We provide answer below and we have updated the manuscript accordingly

Physics Comments

1)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.

Our Reply:We have added the two references suggested.

2)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.

Our Reply: The physics of the emerging quantum correlations due to losses is extensively discussed later on in our review, specifically in Sec. 7.2, and in that context we discuss in detail the physics described by these prior works. This discussion is anticipated in Sec. 2, where at the end of the paragraphs on losses we reference that further discussion in a later section. In Sec. 2 we want to focus only on experimental facts and platform. As such, while we have made a change to add a sentence saying that the intriguing physics associated to losses is discussed later on, we did not feel it appropriate to move this entire discussion into Sec.2.

3)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

Our Reply: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. Both papers were already present in the list of references. While the PRA paper has been introduced in the context of early works on two-body losses and Zeno effect, one of its key results is to discuss the possibility of solving the non-Hermitian Lieb-Liniger model with the Bethe Ansatz. It is cited with a list of references where Bethe-Ansatz-related techniques are employed for the study of open quantum systems. The NJP paper is extensively discussed in Sec. 7.2.2 as it is the basis for the discussion on the Zeno effect. We would prefer not to add these references here as we feel they do not fit well in the discussion at Page 36. We have however added relevant references on Page 36 concerning the Bose-Hubbard model and thank the Referee for pointing this out.

4)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.

Our Reply:We have added a small paragraph to mention these works.

5)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.

Our Reply:We have added references to the earlier work by Plenio, Huelga, and Chin. We note however that there is an important difference between the 2020s work by Kollath and this earlier work. The “chain mapping” approach by Plenio, Huelga and Chin considers the case where the satellite sites in the one-to-many configuration are bosonic modes: i.e. it is motivated by considering the behaviour of an impurity in an environment of harmonic oscillators. The ability to map exactly to an alternative topology, such as the chain, results from the linearity of such a set of harmonic oscillators. In contrast the recent work by Kollath is for the case where the satellite sites are spins, and thus not linear. As such, there is no exact mapping to a nearest neighbour chain model in this case, and instead a more complex process is required involving swap gates. Given the distinction of these two cases we have kept discussion of both, but clarified the difference between them. Regarding the link to DMRG, in response to comments from the other referee we have reorganised the discussion of MPS, and we have now added reference to DMRG in the context of closed systems, as well as related concepts for the density matrix approach.

Style and presentation comments:

7) 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.

Our Reply:We have improved whenever possible the writing and fixed the above mentioned typos.

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

Our Reply:We have fixed this typo.

9)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

Our Reply:We have added this Reference together with Ref. 59.

10)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.

Our Reply:We have added a comment at the end of that paragraph to clarify this.

11)- 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 subtlety.

Our Reply:We agree with the Referee, we have moved and merged the paragraph previously below Eq.(61) with the following subsection.

12)- 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).

Our Reply:We have clarified the definition around Eq. (121)

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

Our Reply:We have fixed this typo.

14)- 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.

Our Reply:We have clarified this ambiguity, which, as now discussed, is ultimately a matter of convention.

15)- 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.

Our Reply:We have updated the Figure (now Figure 33) and the caption to clarify the meaning of the phase boundary

Reply to Referee 2

We thank the Referee for their time in reviewing our manuscript, as well as for the raised questions.

1)Regarding adding additional section on perturbation theory:Splitting the Liouvillian in a dominant and perturbative terms can generally reduce the complexity of studying Liouvillian dynamics, generally down to the complexity of the dominant term. A most naturally studied limit is the limit of weak openness, addressed generically in Sci. Rep. 4, 4887 (2014), PRB 97, 024302 (2018); PRE 101, 042116 (2020). Examples of usage are probably numerious, e.g., PRB 97, 134301 (2018). PRL 121 (26), 267603 (2018), so maybe hard to cite. Of course, this approach can be used also for other limiting cases and is thus quite generic. https://www.nature.com/articles/srep04887, https://journals.aps.org/prb/abstract/10.1103/PhysRevB.97.024302

Our Reply:We have added a new section 4.3 “Perturbative Expansion” and included the relevant references.

2)p.31, discussion after Eq.(20): A recent paper by Mori and Shiraii, PRL 125, 230604 (2020), pointed out that the Liouvillian gap does not necessarily give the slowest relaxation time to the steady state. Although a general understanding of when this happens is, to my knowledge, not yet there, it might be worth pointing out that there can be exceptions. https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.125.230604

Our Reply:We have added a comment on this point and a reference to this work.

3)- page 32: "As discussed in Sec. 1.3, when a system is coupled to a single environment "→was probably meant single thermal environment?

Our Reply:We have updated this sentence.

4)- page 34: "As noted in the introduction, in this review, we will focus on Markovian (i.e., weak coupling) models. Moreover, since we focus on driven-dissipative systems and other nonequilibrium states, we will not generically expect to reach thermal equilibrium. As such, the rest of our discussion will focus on models with local dissipation."→ I disagree since any chaotic Hamiltonian that is weakly coupled to baths in bulk will relax to a density matrix that can be approximated with a thermal state, see works PRB 97, 024302 (2018), PRE 101, 042116 (2020) and as actual solutions PRL 121, 267603 (2018), PRB 97, 134301 (2018), PRL 125, 116601 (2020). In the limit of infinitesimal openness chaotic system relaxes to a thermal state. Maybe the point you want to make is that for obtaining exactly thermal state at finite openness one need to couple to a thermal bath?

Our Reply:The systems we consider are driven and dissipative due to the presence of multiple Markovian baths with a finite dissipation rate, and thus out of equilibrium, i.e. violating fluctuation dissipation-theorem in general. As such, the system density matrix does not in general reach Gibbs equilibrium. This does not mean that in certain parameter regimes an effective equilibrium description cannot be efficient or valid, as in the works mentioned by the Referee. We have added the relevant works to our reference list.

5)- section 3.1.5:I think that under examples of models, it might be worth mentioning coupling to boundary Lindblads that induce a weak current in the system, since this is a rather generic way to probe the transport properties of the bulk model. Crucial references: PRL 106, 220601 (2011), PRB 99, 035143 (2019)

Our Reply:As we have mentioned in the “Summary & Outlook” section, we explicitly disregard boundary-driven settings since they are already covered by other excellent reviews. For this reason we did not include any specific boundary-driven model in Section 3.1.5

6)- page 47, end of Sec. 3:perhaps mention that one physical situation where the no-click limit is applicable is when dealing with condensates with an already macroscopic number of particles/excitation present, so adding/removing one is not crucial.

Our Reply:We do not know of any work to reference which could support such a statement, which a priori seems non-trivial. For this reason we prefer not to comment on this point.

7)- p. 48:under neural network approaches, I would add PRL 127, 230501 (2021) using neural network as a way to approximate the POVM representation of the density matrix. This seems to be a rather promising and generically applicable approach.

Our Reply:We note that this comment seems to relate to a later section, 4.5.3, not any discussion on page 48. Another similar comment by the referee, comment 12, appears below relating to section 4.5.3. We have included the suggested reference in section 4.5.3, and we respond to this comment further below.

8)-p.51:I would possibly add additional references: PRL 126 (24), 240403, arXiv:2406.12695, Phys. Rev. Lett. 124, 160403 (2020), Phys. Rev. Lett. 110, 047201 (2013)

Our Reply:We have added the relevant literature to our reference list, focusing in particular on bulk dissipation rather than boundary (see point 5).

9)- p.62: misprint Θi,eff(t) → Φi,eff(t)

Our Reply:We have fixed this error.

10)-p.60 In that case, I think it would be fair cite other review on driven-dissipative DMFT, e.g. Rev. Mod. Phys. 86, 779 (2014) and arxiv:2310.05201 It might also be useful for students to know that in this framework a very common approach to opennes is to consider a quadratic bath which is integrated out and act as a new term in the effective action. The advantage of such approach is that the bath is non-Markovian and for quadratic baths the procedure is exact. For interacting bath weak coupling expansion is typically considered. Examples of such approachs are numerous and here we only suggest few : https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.110.126401, https://journals.aps.org/prb/abstract/10.1103/PhysRevB.102.165136 , https://journals.aps.org/prb/abstract/10.1103/PhysRevB.106.125123

Our Reply:We have added citations to the nonequilibrium DMFT review and mentioned the linear coupling to quadratic baths in Sect. 4.5.3, adding the suggested references and few more.

11)- p.66: → I would call section 4.5.2 Tensor network methods since the density matrix can also be represented as a matrix-product operator, therefore MPS is a bit misleading.

Our Reply:We have followed the referee’s advice and renamed the section. We have also reorganised the section to address the comments below, and as such we now make clear that the start of the section focuses on matrix product states, while the later part discusses tensor networks more generally.

→ I would mention that using tensor network approaches is very well suited to model open systems (better than for closed systems) since the operator entanglement entropy is naturally bounded by the coupling to the environment. That is the fundamental reason why tensor networks is one of the most popular approaches in the field. While this is common wisedom, it has been addressed in Quantum 4, 318 (2020).

Our Reply:We have added an extended discussion of this point, in the (new) subsection about application of tensor network methods to open quantum systems. We have cited the suggested work and other related work. We thank the referee for suggesting we make clear this important point.

→ one of the most generally used approaches to model open systems with tensor networks is vectorization of density matrix and using the usual methods for states, e.g., TEBD. This possibility is not mentioned at all, even thought it is generically more efficient than sampling over trajectories for state where bond dimension is not bounded by the dissipator. I would strongly suggest introducing it.

Our Reply:As noted above, we have restructured and extended this discussion. In our previous version we had only briefly discussed the different ways the density matrix can be written as a matrix product state and so the significance of vectorisation was not made clear. We now discuss vectorised MPS approaches, MPO approaches, and purification approaches to representing density matrices, and we distinguish these more clearly from the discussion of quantum trajectory approaches.

We have also added an extended discussion on the bond dimension required for different approaches, extending our previous brief discussion on the relation of unravelling to entanglement.

→ Alternative is to latter is also do DMRG-like optimization for the steady state, see PRL 114, 220601 (2015).

Our Reply:In our new extended discussion we now discuss the TEBD and variational algorithm separately, and refer to the suggested work and other relevant work.

12)- p.69: → Figure 17: I think that this schematics is not very informative. For example, the sketch in Hartmann&Carleo [279] is more informative because it makes it more explicit that some hidden neurons connect bra degrees of freedom, some ket DOF, and some connect both (making it a mixed state)

Our Reply:We have revised the figure and caption (now Fig. 19). Comparing our figure to that in the work cited, the structure is very similar, the primary difference is in whether the network is laid out in one row, or in a U shape. It is not clear to us that the different arrangement necessarily makes the role of bra and ket degrees of freedom any more clear. We have however revised the figure (added labels, increased size of l, r index subscripts) and extended the caption with the aim of making this specific point more clear.

→ An alternative to RBM representation is to use a neural network to represent the POVM distribution corresponding to the density matrix, PRL 127 (23), 230501. In this case, one can play with the dept and architecture of the NN to achieve better expressivity. Currently, the neural network architecture doesn't ensure the physical properties of the state. Some other ansätze tried to mend for that, e.g., arXiv:2206.13488

Our Reply:We have added a discussion of this to the end of the RBM section, citing these works and other relevant work on the POVM approach.

13)- p.91:maybe add a reference to Science 383 (6689), 1332-1337, where they implemented system qubits coupled to ancillary ones with a superconducting circuit for the purpose of dissipative cooling.https://www.science.org/doi/abs/10.1126/science.adh9932

Our Reply:We have added a citation to this work.

14)- p. 114, Sec. 7.1.1 would be fair to cite Marko Žnidarič J. Stat. Mech. (2010) L05002 because was before ref 551 and has discussed diffusion due to noise in boundary driven setup https://iopscience.iop.org/article/10.1088/1742-5468/2010/05/L05002

Our Reply:We have added a citation to this work.

15)- p. 117, Eq.(192) t_H → J for consistency with the text

Our Reply:We have modified J→t_H in the text for consistency with the rest of our notation.

16)-p. 125 In addition to [615, 616, 626], optionally mention realization with trapped ions PRR 3, 033142 (2021), quantum computers arXiv:2406.17033 and quenches in Kitaev chain PRL 129 220602 (2022)

Our Reply:We have added a citation to these works.

Section 1: Small changes/reference added (in Response to Referee1)

Section 2 (Experimental Platforms): small changes in the text and added relevant references in Sec. 2.2 and 2.3 (in Response to Referee 1)

Section 3 (Theoretical Frameworks): clarifications and small changes in Sec. 3.1 (Lindbladian Spectrum), Sec 3.2 (Schwinger-Keldysh) and Sect 3.3 (quantum trajectories) – (In response to Referee1)

Section 4 (OVerview Theory methods):Added subsection 4.3 about perturbation theory, clarifications and small changes in 4.5. Major changes in subsection 4.6.2 about tensor networks and MPS (in response to both Referees); Added Figure 16 and Figure 17 on MPS in 1d and 2d. Changes in Subsection 4.6.3 on Neural Network Quantum States for open systems and POVM (in response to Referee 2). Added Figure 19. Clarification in Sec. 4.6.5 on Cumulant Expansion.

Section 5 (State Preparation):minor Changes

Section 6(Dissipative Phases Transitions): updated discussion on dissipative phase transitions in Sec 6.1 (in response to Referee 1). Updated discussion on OPEN DMFT (Section 6.2.2) in Response to Referee 2. Figure 33 Updated (in response to Referee 1).

Section 7: Minor Changes/references added

Section 8 (Monitored systems) – Minor changes//references added