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

Dear Editor,

We would like to thank you for your work and the referees for their valuable comments. In the new version of the manuscript we addressed all their criticisms. Below you can find our reply to their reports.

Sincerely Yours,

Vincenzo Alba Federico Carollo

REPLY TO REFEREES

REFEREE I

R: 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.

A: We thank the referee for considering our work important and timely.

R: 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.

A: We regret that in the introduction we did not manage to motivate enough our choice of the fermionic negativity. As stressed now in the introduction the main reason to consider the fermionic negativity is its calculability. Indeed, while the computational cost to calculate the standard negativity is exponential in the system size, the one to compute the fermionic negativity is only polynomial because the fermionic negativiy is computed from the two-point correlation function. As for the differences between the two negativities, we now stress in the introduction that the two quantities share the same good'' properties of entanglement measures for mixed states. So in conclusion, the both the negativities can be used to quantify quantum correlations in mixed states. This is also supported by the fact that for unitary dynamics in closed quantum systems the two negativities become the same in the scaling limit.

R: 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

A: We thank the referee for pointing to us these papers. We are sorry that we missed them in the first version of the manuscript. Indeed, these papers are important, especially because they study the moments of the partial transpose density matrix, and in a way they motivated the introduction of the fermionic logarithmic negativity. We now mention these important papers in the manuscript.

R: 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?

A: We regret that we were not sufficiently clear about this point in the manuscript. The results presented in sec. 4 can be considered original because the derivation that we provide is different from that given in our previous work. In fact, the main virtue of the derivation in sec. 4 is that it is ab initio. In contrast, the result in Ref. [7] were heavily based on the final formula (61) of Ref. [84]. To apply the result of Ref. [84] one has to notice that our initial state, i.e., the Neel state, is not translational invariant. On the other hand, the results of Ref. [84] hold for quenches in the XY chain from generic translation invariant initial states. This difficulty can be overcome, as discussed in Ref. [7], by transforming the initial state and the Hamiltonian with a local unitary operator. The problem now becomes that of the dynamics from a ferromagnetic state (which is translation invariant) with a rotated'' Hamiltonian, which is still in the class of XY models treated in Ref. [84]. Moreover, the rotation is not crucial because the Renyi entropies are not affected by the unitary transformation. However, another difficulty is that the results of Ref. [84] hold for the even moments of the Majorana fermionic correlation matrix, because in the unitary settings the odd ones are zero. This is not the case in the presence of gain and loss dissipation. To overcome this issue in Ref. [7] we restricted ourselves to the case of balanced gain and losses. In that case the even moments of the fermionic correlator are sufficient to obtain the entropies. This allowed us to obtain our conjecture for the entropies (Eq. (13)) for \gamma^+=\gamma^-. From (13) it was straightforward to conjecture the result for arbitrary \gamma^+,\gamma^-.

Now, the calculation of the fermionic negativity is much more involved than that of the Renyi entropies and applying directly the results of Ref. [84] is not as easy. For this reason, and as a warm-up before attacking the negativity, we decided to rederive the result for the entropies. We believe that the derivation is instructive, also considering the it takes only few pages and it can be easily followed by the reader interested in generalizing our results.

This is now better presented in the new version of the manuscript. For instance, now we write explicitly at the beginning of the section that our derivation is ab initio, although, of course, our approach is similar to Ref. [84], which pioneered this type of calculations. We also stress some aspects that are different as compared with the derivation of Ref. [84]. For instance, the fact that the initial state is two-site translation invariant implies that within the stationary phase approximation the stationary point is not unique but there is a proliferation of stationary points, due to the invariance by a \pi shift.

R: 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 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.

A: We thank the referee for this important comment. We now extended significantly the discuss of the physical origin and consequences of our formula. In particular, we now compare the dynamics of negativity in the presence dissipation with the results of Ref. [72]. In the new paragraph we stress that in the presence of dissipation the linear growth at short times is damped exponentially although the negativity exhibits the typical rise and fall'' dynamics as in the unitary case. We also stress, again, that the negativity content of the quasiparticles is not the same as the Renyi mutual information. This is in contrast with the unitary case. Following the suggestion of the referee we also added a new figure for the case of balanced gain and loss dissipation. We also discuss the structure of the negativity content. In fact we comment on the possibility that the negativity content of the quasiparticles could be obtained from an effective few-states mixed density matrix for the two subsystems.

R: 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?

A: As we stressed in the previous point it is expected that the negativity content of the quasiparticles can be obtained from an effective few-states mixed density matrix for the two subsystems. The size of this density matrix is expected to depend on the number of dissipative processes that can affect the two subsystems. One could imagine that for independent gain and losses there should be a 4x4 matrix. In the case of balanced gain and losses one could expected that gain and losses can be treated as a single dissipative process. This would lead to a reduction of the effective density matrix. This suggests that the fact that for balanced gain and losses we obtain the simple expression (90) is accidental'', and a similar formula for generic gain and losses is not expected.

R: 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.

A: We thank the referee for this suggestion. We moved some of the technical derivations in the Appendix. We kept the derivation of the moments Tr(G^+G^-)^n in section 5.1. In section 5.2 we report the main formulas for the moments with defects insertions, which we believe are interesting. The derivation is reported in Appendix A.

R: 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.

A: We thank the referee for the remark. We now put more details on the quench protocol.

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

A: We corrected that.

R: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.

A: We modified the captions of the figures accordingly.

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

R: Fixed.

R: 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)

A: We thank the referee for this comment. We fixed the full stops/comas.

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

A: We modified the manuscript accordingly.

REFEREE II

R: 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.

A: We thank the referee for the positive assessment. In the new version of the manuscript we addressed all the criticisms of the referee.

R: 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.

A: We thank the referee for the comment. We agree that the sentence might sound contradictory. We meant that while for pure states there are several computable measures of entanglement, such as von Neumann and Renyi entropies, for mixed states the negativity is the only one that is computable. Moreover, as we write in the introduction the calculation of the negativity is a challenging task except for free bosons. Indeed, for free fermions, only the fermionic negativity can be computed effectively. In numerical simulations with DMRG the algorithms to compute the negativity are not as effective as the ones for the entropies. Indeed, the typical computational cost to compute the negativity scales as \chi^6 with \chi the bond dimension. This has to be compared with the cost \propto\chi^3 for the entropies. Finally, there is no efficient quantum Monte Carlo algorithm to compute the negativity, whereas the Renyi entropies can be computed effectively with QMC. We modified the sentence in question as

"While for bipartite quantum systems in a pure state several computable quantum information motivated measures can be used to identify entanglement, this is more challenging for mixed-state systems"

R: (II) 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.

A: We thank the referee for this comment. Indeed, the paragraph was not precise. We modified as

For generic interacting systems 
it is quite challenging to build a quasiparticle picture to describe the full-time 
dynamics of R\'enyi entropies, although 
their value in the steady state can be determined. 
Moreover, recent exact results for quenches in the so-called rule 
$54$ chain~\cite{klobas2021entanglement}, which is a ``minimal model'' 
for  interacting integrable systems, suggest that R\'enyi entropies violate 
the quasiparticle picture paradigm. These results motivated a 
conjecture for the growth with time of the R\'enyi entropies in 
generic interacting integrable systems~\cite{bertini2022growth}. 
However, Ref.~\cite{bertini2022quantum} showed that in the early-time regime and for 
contiguous subsystems the relation between Renyi-$1/2$ mutual 
information and logarithmic negativity put forward in Ref.~\cite{alba2019quantum} 
still holds for  any local 
quantum circuit, and therefore also for interacting integrable 
systems. Similar results were obtained in CFTs~\cite{kudler2021the}.

R: "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.

A: We modified the text to be more precise, see previous point.

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

A: We thank the referee for the comment. We now write that L is even below Eq. (4).

R: 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>.

A: In the new version of the manuscript we implemented the suggestion of the referee.

R: "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.

A: We thank the referee for this comment. The reason of the confusion is that we had in mind of defining with the tilde the correlation function for the case without dissipation and without the tilde the correlation in the presence of both dissipative and unitary dynamics. To make this clear we now use the tilde in section 2 and the notation without the tilde in 2.1.

R: 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?

A: We removed the sentence in question.

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

A: We thank the referee for the comment. We now added a sentence to make explicit that (12) is obtained from (11) just using the definition of the fermionic correlation function, the dynamics of the density matrix, and Wick's theorem to perform the trace. We also add a Ref. [33] where the formula is reported and further explained.

R: 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.

A: The referee is indeed right that as it is the sentence is not very clear. We added few sentences to explain how (18) can be derived. The key point is that in the limit of small dissipation rates, (18) is to be expected if one assumes that the eigenvalue and the eigenvector of the Liouvillian differ from those of the Hamiltonian by terms that are linear in the dissipation rates.

R: 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.

A: We corrected according to the referee suggestion.

R: 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?

A: We thank the referee for the comment. First, the fermionic and the standard negativity do not coincide. However, they both can be taken as bona fide entanglement measure, as we now write explicitly in the introduction. The meaning of this fact is discussed in Ref. [21].

R: 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.

A: We modified the notation. Now the correlator for the case without dissipation is denoted by a tilde. We now refer to (12) for the correlator that is needed to construct the negativity.

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

A: The referee is right that C_A is the restricted correlation matrix. This is now explicitly stated.

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

A: In the new version of the manuscript we added a few remarks on the meaning of the two terms in (29). We also cite a new reference [51] where that formula is derived.

R: 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).

A: We thank the referee. We corrected the sentence.

R: 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.

A: We agree with the referee that this is not a discussion. However, it is not just a reiteration of a statement that is present in section 5.3. Thus, we prefer to keep it to make it clear the regime of validity of the result. We rephrased the final paragraph to make it clear that it not as a discussion.