Logarithmic negativity in out-of-equilibrium open free-fermion chains: An exactly solvable case

1- timely topic

2-interesting quantity considered

1- presentation focused on the technical aspects and not on the physics of the results

2-poor discussion of the results and comparison with previous literature

The authors have investigated the fermionic logarithmic negativity (obtained from the partial time-reversed reduced density matrix,
not from the standard partial transposition) for a tight binding chain on the line after a a global quantum quench from the Need state.
The system is open, hence dissipation occurs an this is main novelty with respect to the existing literature.
The subsystem is made by the union of two disjoint intervals and the partial time reversal for one of them is considered.
They mainly work in the weakly-dissipative hydrodynamic limit. In this regime, analytic expressions can be found by extending
existing methods developed for the closed systems.

The authors extends to this quantity the analyses they did earlier, in other papers, for the entanglement entropies in a similar setup.
The main result of the manuscript under review is described in sec. 5 (see e.g. eq. (87)) and the corresponding numerical checks are discussed in sec. 6.

Understanding entanglement in open quantum systems is important and timely.
The results of the manuscript are interesting along this line and nicely reported, hence I support the publication of this paper in SciPost.

The major issues to address are

(A) As for the motivations, the authors should motivate why they consider the so-called fermionic negativity and not the standard one
for free fermions. Are there physical reasons or just technical ones? Furthermore,
for the non-experts, the authors should remark the differences between these two quantities.

(B) in the text of pag. 7, three lines below eq. (23) the authors claim that the standard fermionic negativity “cannot be easily calculated, not even for free fermion models”.
Beside the technical difficulties, analytic results have been obtained in the literature (e.g. 1508.00811, 1503.09114, 1601.00678) that the authors could cite

(C) In sec. 4 the authors claim that their previous result for the entanglement entropies is obtained here in a different way.
However, it is not clear which is the novelty of this derivation and why it should be preferred to the previous one.
The derivation is presented as a list of technical steps without any discussion or comparison with the previous analysis.
Which are the steps of the derivation that must be changed with respect to the procedure followed in ref. [81]?
Where the procedure becomes simpler in the limiting regime of closed system?

(D) The main result is presented at pag 19. Just a technical description of the formulas is given.
It is highly recomended to enlarge this discussion by including considerations about the physics of the result.
For instance, a comparison can be made with existing results about the temporal evolution of the negativity,
e.g. against ref. [72] where the typical “rise and fall” behaviour shown in fig. 4 for the negativity has been first obtained for the standard negativity
(for closed systems) or against the temporal evolution of the fermionic one for closed systems.
It could be useful also to have a plot like the the one in fig. 4 in the case of balanced loss/gain dissipation
showing what happens in the limiting regime when $\gamma^+ = \gamma^-$ goes to zero and the result without dissipation is recovered.

(E) an important consequence of the main result is the relation (90), which does not hold in the generic case, as the authors properly remark.
Why a similar relation cannot be found for the generic unbalanced case?
What are the difficulties? Just technically complicated expressions to deal with or more conceptual obstacles occur?

(F) As overall remark, I found the presentation too technical. It seems to me that many technical details could have been appendix material.
During the revision, the authors might consider the possibility to move some derivations in the appendix, in order to focus the attention of the reader
on the result and on its interpretation, and not on the steps of its derivation.


Minor issues and typos are the following

(I) the authors should clarify better the evolution dynamics.
For instance below eq. (3) they claim “non equilibrium dynamics after the quench from the fermionic Neel state” but it is not clear
what is the final stage of the quench.

(II) in eq. (20) the subindex X could be W, to be consistent with the previous text.

(III) in the caption of figs. 4-5-6-7, it would be useful to indicate the equations of the text employed for the analytic curves.

(IV) typo in eq. (56): parenthesis are missing in the l.h.s., in order to be consistent e.g. with eq. (53).

(V) all the full stops/comas at the end of the formulas should be revised
(see e.g. eps. (37), (44), (45), (52), (62), (64), (74), (76) and presumably some other ones)

(VI) in eq. (90) it could be helpful to add the text “ for $gamma^+ = gamma^-$ ” within the equation , as done in eq. (89)

The authors study the dynamics of logarithmic negativity in an open free-fermion chain. They consider a tight-binding model with the addition of on-site dissipation. After introducing the model, and the so-called "fermionic negativity" (a quantity closely related to negativity that is easier to calculate in free-fermionic systems), they derive expressions describing the dynamics of Renyi entropies and negativity, valid in the special scaling limit of subsystem sizes and time going to infinity with fixed ratios, which they refer to as the "hydrodynamic limit", while the dissipative rates need to scale as the inverse of the subsystem size (referred to as "weakly-dissipative" limit). Interestingly, they find that the usual relation between the Renyi-1/2 mutual information and negativity ceases to hold. Surprisingly, in certain limits they recover an analogous relation between negativity and Renyi-2 mutual information. They also provide numerical data that supports the analytical predictions.

I believe that the main results of the paper are interesting and deserve publication, but not in the current form. Derivations are hard to follow up to the point of the paper being almost unreadable. I am aware that this is a calculation-heavy paper, and is therefore technical in nature. However, the authors should make a stronger effort to make the paper more accessible. I propose the authors try to clean the calculations up, add explanations where relevant, and maybe move some of the more technical parts to an appendix. Moreover, the sections preceding the derivation are missing some more intuitive explanation of the physics discussed there. It feels like a recount of previous results by the authors without enough context to make the paper self contained. I do not expect the authors to rederive all the previous results leading to the calculation in Sec. 5, but it would be nice to get an idea of where most of the statements come from without having to be thouroughly familiar with an extensive list of literature.

Furthermore, I find that the writing is at certain points sloppy with some statements that are misleading or even plainly false. Below I'm attaching a list of examples that I found.

1st paragraph:
"While for bipartite quantum systems in a pure state several quantum information motivated measures can be used to identify entanglement [1-4], this is not the case if the state of the full system is mixed." <- This seems to be incorrect: the authors in the next paragraph introduce a measure for mixed-state entanglement, which is also motivated by quantum information.

Page 2, 2nd paragraph:
"For interacting integrable systems no results are available." <- This is again not true. Ref. [80] suggests that in the early-time regime and for contiguous tripartition this relation between Renyi-1/2 mutual information and logarithmic negativity holds exactly (i.e. not only in the scaling limit). This result holds for any local quantum circuit, and therefore also for interacting integrable systems. See also Ref. [75] for a similar claim in the case of CFTs.

"Despite this scenario, it is possible to obtain analytically the dynamics of the logarithmic negativity in the rule 54 chain, and it is in agreement with Ref. [23]." <- Again, this is misleading. As far as I'm aware there is no explicit calculation for Rule 54. The only result that could come close to that is the general statement of Ref. [80] that holds for *all* local unitary circuits (see above), but this is only in agreement with Ref. [23] in the early-time regime.

Sec. 2: I am assuming that L is even. It could be useful to point it out explicitly.

Eq. (5): It could be useful to define what kind of expectation value it is, i.e.
C_{jl}:=<c_j c_l^{\dagger}>=<\Psi_0|c_j^{\dagger}c_l|\Psi_0>.

"time-dependent correlation function \tilde{C}_{jl}(t) [cf. (5)] <- This doesn't explain at all how this is defined. From the context it is clear that this refers to <\Psi(t)|c_j c_l^{\dagger}|\Psi(t)>, and therefore there is no good reason why both zero-time and t>0 correlation functions can't be defined with the same equation. Moreover, I do not understand why one of them is denoted by C and the other \tilde{C} - especially since in the next section authors introduce C which is the correlation function in the presence of dissipation.

Line after eq. (11):
"Here, {x,y}=xy+yx is the anticommutator" <- Why is the anticommutator introduced here, if it is already used on the previous page?

Eq. (11-12):
It could be useful to add aditional details on how (12) follows from (11).

Eq. (18):
Where does this follow from? Can a few words be added to support this? I realize it is an old result, but some additional explanation could be useful.

Eq. (20) & line before that:
Maybe the same letter can be used for the subsystem, i.e. instead of W in the text and X in the equation it could both be X or both W.

Sec. 3: Could a few words be added to say whether or not fermionic negativity and usual negativity are equivalent in some sense? How do I know that they measure the same thing?

Line before eq. (25):
"The central object is the fermionic correlation matrix C_{jl} (cf. (5)).". <- Are the authors referring to C of dissipative or non-dissipative case? They are using C as in the dissipative definition, but they are referring to the non-dissipative quantity. I am assuming it's the dissipative one, so reference to eq. (5) is probably wrong.

Line before Eq. (28):
What is C_A? I assume it's C restricted to the subsystem A, but it's never defined.

Eq. (29):
How do we get this expression? A few words of explanation would be nice.

Last paragraph on pg. 8:
"Very recently, Eq. (30) has been verified for quenches in the rule 54 chain [80]." <- This is wrong, Ref. [80] does not deal with Rule 54, but with general circuits (see a remark above).

Last paragraph before Sec. 6:
This is not a "discussion" of the regime of validity of (87), the paragraph is just reiterating a statement from before.