On the generality of symmetry breaking and dissipative freezing in quantum trajectories

1 - Well-structured and accessible manuscript, suitable for a wide readership.
2- Timely topic
3- Nice set of examples to help understanding and giving concretness for the general results.

1- Lack of a significant breakthrough on a previously-identified and long-standing research discovery (i.e., dissipative freezing)

I read carefully all the versions of the manuscripts and the previous referee's comments. I find the topic interesting and the results worthy of publication and I acknowledge the improvements made by the authors, but I do stick to the opinion of Referee 2 on 2022-5-19 that the manuscript will be more suited for SciPost Physics Core.

Indeed, I do not really see how the present manuscript meets at least one of the "Expectations" criteria of SciPost Physics:

1. There is no "groundbreaking theoretical/experimental/computational discovery", since dissipative freezing was already discovered in a previous work.

2. I do not find that the manuscript "presents a breakthrough on a previously-identified and long-standing research stumbling block" (i.e. dissipative freezing). Indeed, even if Ref [24] was only presenting dissipative freezing in a particular case, it was already clear that the presence of a strong symmetry was a key ingredient. In the present manuscript, the authors provide significant mathematical and physical insight into why freezing will emerge, but I would not qualify it as a breakthrough. It is rather an elaboration and a generalisation of a previously found result.

3. I do not find either that this manuscript "opens a new pathway in an existing or a new research direction, with clear potential for multipronged follow-up work". The manuscript provides a way to identify better whether or not a system undergoes dissipative freezing, but I would not qualify it as a new research direction: I do not find the manuscript impactful and helpful enough for the scientific community for guiding researchers in new research directions. How could one use the results of this paper for this purpose ?

4. I do not find either that it "provides a novel and synergetic link between different research areas". They identify a direct connection between rigorous mathematical results on ergodicity in open systems and dissipative freezing, but again there was already the idea in [24] thet dissipative freezing was an non-ergodic phenomenon. Also, the synergy between different fields is unclear.

Hence, while the present manuscript constitutes a significant elaboration on the mechanisms underlying dissipative freezing, I do not think it satisfies the publication criteria of SciPost Physics and how it would open significant research directions for the readership. However, I find that it clearly meets more adequately all the acceptance criteria of SciPost Physics Core:

1. "Address an important (set of) problem(s) in the field using appropriate methods with an above-the-norm degree of originality;"
2. "Detail one or more new research results significantly advancing current knowledge and understanding of the field."

and I would thus recommend it for publications in SciPost Physics Core, except if the authors can revise their manuscript so to provide materials that convince more strongly on how their manuscript meets the acceptance criteria of SciPost Physics.

- I suggest (as Referee 1 on 2022-8-9) a typo check.

-In the caption of Figure 2, it is said \gamma = 4 \omega. To make clearer the connection with Eq. (1), the number M of jump operators and/or the index j of \gamma should be specified.

- In the example 3.3, the authors work with qudits but below Eq. (29) mentions the "spin-3/2" operator. The dimension of the qudits used in the example should be mentioned more clearly.

1. Well structured and accessible as only minimal knowledge of linear algebra is required.
2. Convincing examples demonstrate dissipative freezing and exceptions to it.

1. Lacks precision and rigor in its arguments and conclusions

Dissipative freezing corresponds to a recently observed property of the time-evolution of an open system with a strong symmetry to asymptotically project each quantum trajectory into a single subspace of the strong symmetry, thereby spontaneously breaking the latter at the level of individual trajectories.
The manuscript attempts to provide a broader view on this phenomenon that should be accessible to most physicists while also attempting to elevate the discussion previously constrained to specific examples to a more formal level. Unfortunately, the manuscript falls short of its ambitions, ending up as neither fish nor fowl.
The point where this becomes most painfully clear is the box on page 5, which is unsuitable as a mathematical definition but also provides little physical insight as no concise conditions for exceptions are provided.
The level of the analysis of dissipative freezing presented in the manuscript, nevertheless, significantly exceeds the existing literature. I am therefore generally inclined to recommend publication if the following minor points can be clarified/corrected or improved:

1. Content:

In the introduction, I would appreciate a paragraph on the physical relevance of dissipative freezing. It is discussed later, but in my opinion should also be emphasized in the introduction.

Below equation (14) there appears to be a typo as neither $\alpha$ nor $\alpha’$ are defined in this context. I therefore suggest to replace “is largest in $\alpha$ or $\alpha'$” by “is larger in $\alpha_1$ or $\alpha_2$”.

At the beginning of page 10, it doesn’t seem necessary, nor guaranteed by the definition of A that the steady state of each symmetry sector is unique. To be explicit, A might not split the Hilbert space in the maximal number of subspaces.

At the beginning of page 11 I fail to follow the argument
“if $\tilde{p}^{(i)}(\alpha_{1}, t)\tilde{p}^{(i)}(\alpha_{2}, t) \ll 1$ we have that one of $p^{(i)}(\alpha_{1},t)$ or $p^{(i)}(\alpha_{2}, t)$ will be on the order of $\epsilon^{2}$”
In this context it might also help to introduce the symbol $t_f$ for the freeze-time.

In section 2.5 $\Delta^{(\alpha_{1}, \alpha_{2})} = 0 \ \forall \alpha_{1} \neq \alpha_{2}$ should read $\Delta^{(\alpha_{1}, \alpha_{2})} \neq 0 \ \forall \alpha_{1} \neq \alpha_{2}$. However, as is explained in the following section on inter-sector traceless modes, even this statement is not generally valid.

Conversely in equation (26) one should have $\Delta^{(\alpha_{1}, \alpha_{2})} = 0$. On a more general note, this condition doesn’t seem very useful as, on the one hand, it is not being linked to any properties of the Hamiltonian or Lindblad operators. On the other hand, it is also computationally expensive to check. If possible I would appreciate if the authors code provide a statement to which extend equation (26) may be of use.

The argument on page 17 “If the trajectory is left to evolve over a time-scale sufficiently longer than that associated with the inverse of $\Delta^{(\alpha_{1}, \alpha_{2})}$, then any signatures of freezing should be apparent. If they are not then it is reasonable to assume freezing will not occur and traceless non-decaying modes are present.” is circular as an evolution to $t\sim 1/\Delta^{(\alpha_{1}, \alpha_{2})}$ excludes the case of traceless non-decaying modes characterized by $\Delta^{(\alpha_{1}, \alpha_{2})}$


2. Grammar/Spelling:

While the general level of grammar is very good, in its current form the manuscript is riddled with punctuation errors. Due to their huge number, I will not provide the detailed location, but rather the immediate context:

- far reaching implications which→ far-reaching implications, which

- model at hand an analytical → model at hand, an analytical

- numerical examples, from a range → numerical examples from a rang

- absense → absence

- dissipative freezing we provide a novel → dissipative freezing, we provide a novel

- system symmetries can no longer → system, symmetries can no longer

- also true, if there → also true if there

- Evan’s → Evans or Evans’

- Unravellings which are to higher order → Unravellings which are of a higher order

- symmetry subspace then the phenomenon → symmetry subspace, then the phenomenon

- off-diagonal block we will refer them as → off-diagonal block, we will refer to them as

- states in different subspaces then dissipative → states in different subspaces, then dissipative

- constraint independently → constraint, independently

- of the GSKL equation we can → of the GSKL equation, we can

- this make sense → this makes sense

- jump operators it follows from → jump operators, it follows from

- quantities values → quantities’ values

- the, randomly selected → the randomly selected

- freezing we now → freezing, we now

- our analysis we work → our analysis, we work

- block-diagonal then the Liouvillian → block-diagonal, the Liouvillian

- If this is true then → If this is true, then

- simlarity → similarity

- any, other forms of → any other forms of

- respresent → represent

- is protected from the system’s → are protected from the system’s

- examples we → examples, we

- routine we → routine, we

- As our first example we → As our first example, we

- Lindblad operators which are block → Lindblad operators, which are block

- and in this example these → and, in this example, these

- a certain time the total → a certain time, the total

- the long-time limit the probability → the long-time limit, the probability

- identical, aside → identical, aside

- Importantly this→ Importantly, this

- intuition we have built up → intuition we have built

- all commute there → all commute, therefor
- follows the number → follows that the number

- Notably as $\gamma$ → Notable, as $\gamma$

- is traceless modes → are the traceless modes

- for sufficiently long times we can → for sufficiently long times, we can

- diverges then any given → diverges, then any given

- are not then → are not, then

- much more computationally → a much more computationally

- vanishes then the Liouvillian → vanishes, then the Liouvillian

- posseses → possesses

- exceptions would → exceptions, would

- possesion → possession

- system the bound → system, the bound

- and then general dependence of the the freezing→ and then the general dependence of the freezing

- where the $c^{(i)}_{\alpha, \beta}(t)$ are the coefficients of the $i$th trajectory → where $c^{(i)}_{\alpha, \beta}(t)$ is the coefficient of the $i$th trajectory

- to ensemble average → to ensemble-average

- open systems) the more constraints → open systems), the more constraints

- qubit → qudit