SciPost Submission Page
Steady-state entanglement scaling in open quantum systems: A comparison between several master equations
by Antonio D’Abbruzzo, Davide Rossini, Vittorio Giovannetti, Vincenzo Alba
Submission summary
| Authors (as registered SciPost users): | Vincenzo Alba · Antonio D'Abbruzzo |
| Submission information | |
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| Preprint Link: | scipost_202501_00020v1 (pdf) |
| Date submitted: | 2025-01-12 16:56 |
| Submitted by: | Alba, Vincenzo |
| Submitted to: | SciPost Physics |
| Ontological classification | |
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| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
Abstract
We investigate the scaling of the fermionic logarithmic negativity (FLN) be tween complementary intervals in the steady state of a driven-dissipative tight- binding critical chain, coupled to two thermal reservoirs at its edges. We compare the predictions of three different master equations, namely a global Lindblad equation, the Redfield equation, and the recently proposed univer sal Lindblad equation (ULE). Within the global Lindblad equation approach, the FLN grows logarithmically with the subsystem size $\ell$, for any value of the system-bath coupling and of the bath parameters. This is consistent with the logarithmic scaling of the mutual information analytically demonstrated in [Phys. Rev. B 106, 235149 (2022)]. In the ultraweak-coupling regime, the steady-state FLN obtained from the Redfield equation and the ULE in- creases logarithmically as $\mathscr{E}\propto \widetilde{c} \ln(\ell)$. The prefactor $\widetilde{c}$ is nonuniversal, and it is the same as in the global Lindblad equation. Such behavior holds even when moving to moderately weak coupling and intermediate values of l. However, when venturing beyond this regime, the FLN crosses over to superlogarithmic scaling for both the Redfield equation and the ULE.
Author indications on fulfilling journal expectations
- Provide a novel and synergetic link between different research areas.
- Open a new pathway in an existing or a new research direction, with clear potential for multi-pronged follow-up work
- Detail a groundbreaking theoretical/experimental/computational discovery
- Present a breakthrough on a previously-identified and long-standing research stumbling block
Current status:
Reports on this Submission
Strengths
1) Investigation of the interplay of entanglement and nonequilibrium conditions.
2) Comparison of different quantum master equations in an approachable setup.
See the full report for details.
Weaknesses
1) The scaling of the entanglement negativity cannot be resolved with the system sizes used.
2) The quantum master equations are not compared against exact results, leaving open the question of which one is more accurate.
See the full report for details.
Report
The manuscript addresses quantum information measures in nonequilibrium systems, two topics which, separately and together, have been at the center of much recent interest. The specific system is a 1D fermionic lattice coupled to two reservoirs at different temperature, which are taken into account using three different formulations of the quantum master equation. The system is quadratic, hence allows solution with moderate numerical effort. The solution is used to find the fermionic entanglement negativity between two subsystems and its scaling with their size. It is found that at weak system-bath coupling all the methods reasonably agree (as they should) and give rise to similar logarithmic negativity scaling, in accordance with previous studies. At stronger couplings the results of the different master equations could be quite different, and for some of them there might be deviations from logarithmic scaling, which are however not well resolved for the system sizes used.
While the work is technically correct as far as I can judge, as furthermore is well written and presents an interesting contribution to the quest of understanding these various master equations and their limits of applicability, the work seems to fail short of providing conclusive answers to the questions it poses. In particular:
1) The system sizes used in the work do not allow to conclusively determine whether the negativity scaling indeed deviates from logarithmic. But I see no reason to stop at systems of size ~10^3. Since the numerics involves linear algebra in the system's single-particle space, there should be no problem, based on my experience with similar systems, to reach sizes of order 10^4 on a single-CPU machine; the calculation of each data point should take a couple of hours, which is still within the reasonable range. With a small cluster even larger systems could be approached. This would allow to settle the negativity scaling question.
2) An even more important issue which is left open is which master equation is to be trusted, given that they are all based on approximations which become uncontrolled once the system-bath coupling is not weak. However, the quadratic nature of the system allows for an exact solution of the full system+bath: One may find the scattering states originating in either bath (let me note this would be a bit easier for a system-bath coupling which conserves the fermion number, as opposed to the model used in the manuscript, but this is not essential), populate these scattering states according to the respective distributions, and calculate the resulting system correlation matrix and negativity. All the steps could be formulated as linear algebra problems in the system single-particle space ("integrating out" the baths), hence should be solvable with a similar numerical effort to the master equations. This exact solution will allow to decide whether and which of the master equations provides a good approximation.
Thus, it seems that with moderate effort the work could be extended to give more conclusive answers. I believe this is necessary for the manuscript to deserve publication in SciPost.
Requested changes
1) Extend the numerical calculations to larger system sizes to resolve the negativity scaling.
2) Compare the quantum master equation results to exact ones.
See the full report for details.
Recommendation
Ask for major revision
Strengths
1. The detailed description of how to compute two point correlation matrices from three different master equations for open quantum systems and how one connects to the others.
2. The numerical study of scaling of upper bound of negativity as an entanglement measure for steady states for free fermion systems connected to reservoirs is very timely with the interest in open quantum systems.
3. The difference in behaviour seen for Lindblad equation compared to the Redfield equation and universal Lindblad equation for larger system-bath couplings beyond a certain system size indicates the effect seems to be persistent in thermodynamic limit. This may trigger further works regarding the validity of each method for computing the steady state in different scenarios.
4. Clear statement of the results presented with no over claims.
Weaknesses
1. Unclear as to why only system sizes L~10^3 reached for computing steady state data for free fermion systems even taking into account the open system setup.
2. Lack of interpretation of the results which has been left for future works and no clear conclusion has been drawn regarding the what the observations mean from a physics perspective, it looks like a statement of some numerical results which I feel is inadequate for a journal like Scipost physics.
Report
The work is a timely study of the entanglement in steady states obtained from three different master equations which approximate open quantum system physics. The usage of quadratic models allows the authors to compute an upper bound of negativity for the steady state density matrix using only two point correlation functions. The authors unveil differences in entanglement scaling of steady state with system size for larger system-bath coupling strengths for the different master equations.
I feel in general this is an important numerical study to shed light on the validity of different master equations with a cautionary tale on their usage. However it is hampered by a lack of proper interpretation of the numerical results and certain numerical limitations for which I suggest a revision is in order before publication.
Requested changes
1. At least for Lindblad systems, similar quadratic Hamiltonians (Physical Review B 107 , 184303 (2023)) have been studied for sizes~10^4 for dynamics. Since this is a fully numerical work, state-of-the-art numerical results are better, or a clear explanation why reaching higher system sizes is not possible should be provided.
2. Statements like "The superlogarithmic growth is likely due to the weak-coupling assumption in the Born-Markov approximation that is used to derive both the Redfield and the ULE [1, 48], and it signals the fact that the limit ℓ → ∞ and γ → 0 do not commute.” needs to be more rigorous. Because the one scenario where the authors get logarithmic growth- the Lindblad equation, in my opinion uses the same approximation, so I cannot agree with this statement. The non-commuting behaviour is also present in Lindblad dynamics (see Phys. Rev. B 109, 064311 (2024)).
3. In my opinion, a proper interpretation of the results should be provided, in the abstract and in the main text, to strengthen the quality of the work, currently the authors have only made speculations. It is hinted that the authors expect logarithmic behaviour for entropy from Phys. Rev. B 106, 235149 (2022), but that in itself uses Lindblad equation so I am confused. Or is it due to the conformal theory description for the closed system?
4. Since the quantity computed is the upper bound of negativity, examples with small systems size in the appendix may be provided to give an indication how far from the actual value do the values computed in the manner of the paper are.
5. Questions may arise whether the choice of spectral density affects the results, an intuition regarding that would be helpful for the reader.
6. There are a large number of typos in text, please revise to correct them.
Recommendation
Ask for major revision