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Exact Entanglement in the Driven Quantum Symmetric Simple Exclusion Process
by Denis Bernard, Ludwig Hruza
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Submission summary
| Authors (as registered SciPost users): | Ludwig Hruza |
| Submission information | |
|---|---|
| Preprint Link: | scipost_202306_00041v1 (pdf) |
| Date submitted: | 2023-06-29 16:41 |
| Submitted by: | Hruza, Ludwig |
| Submitted to: | SciPost Physics |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
Abstract
Entanglement properties of driven quantum systems can potentially differ from the equilibrium situation due to long range coherences. We confirm this observation by studying a suitable toy model for mesoscopic transport~: the open quantum symmetric simple exclusion process (QSSEP). We derive exact formulae for its mutual information between different subsystems and show that it satisfies a volume law. Surprisingly, the QSSEP entanglement properties only depend on data related to its transport properties and we suspect that such a relation might hold for more general mesoscopic systems. Exploiting the free probability structure of QSSEP, we obtain these results by developing a new method to determine the eigenvalue spectrum of sub-blocks of random matrices from their so-called local free cumulants -- a mathematical result on its own with potential applications in the theory of random matrices. As an illustration of this method, we show how to compute expectation values of observables in systems satisfying the Eigenstate Thermalization Hypothesis (ETH) from the local free cumulants.
Current status:
Reports on this Submission
Report #2 by Anonymous (Referee 2) on 2023-8-25 (Invited Report)
- Cite as: Anonymous, Report on arXiv:scipost_202306_00041v1, delivered 2023-08-25, doi: 10.21468/SciPost.Report.7708
Report
The authors of this paper compute the mutual information between interval-like subregions in a 1D model of the QSSEP. The motivations as well as the method employed seem to be consistent with the previous publications on the topic. I have however a few remarks:
1. In the introduction, it is suggested that the volume law of mutual entropy is an out-of-equilibrium phenomenon since systems at equilibrium follow an area law at zero temperature, according to ref [7]. It would be helpful to confirm this idea on QSSEP, by singling out the limit of equal reservoir densities, one expects a breakdown of the area law in this limit since the model becomes at equilibrium. It would be constructive to provide insight into how this can occur within the formalism presented in the paper and to refer to relevant existing literature if this limit has already been addressed.
2. It's not clear in the paper why specifically the Reyni entropy of order 2 has been chosen.
3. The methodology employed for the numerical simulation is completely absent.
4. In section 2, the usage of indices belonging to a continuous real interval for a discrete model is a bit confusing and becomes only clear in section 3, Clarifying the relationship between these indices should be considered upon their introduction.
5. In section 2, the conclusion of the mutual information scaling as the volume of the subregion seems to rely on the figure, One can argue that there might be other higher non-linear terms with small coefficients. The precise meaning of the scaling in terms of the size of the subsystem and the total system is not stated clearly enough.
6. There is a missing minus sign in the definition of entanglement entropy in the introduction.
7. In the introduction, it might be better to refer to the area of the boundaries of the subregion, rather than the area of the subregion itself, as it can be confusing for readers who are not in the field.
Requested changes
Addressing the remarks in the report.
Report #1 by Anonymous (Referee 1) on 2023-8-13 (Invited Report)
- Cite as: Anonymous, Report on arXiv:scipost_202306_00041v1, delivered 2023-08-13, doi: 10.21468/SciPost.Report.7649
Report
The title starts with "Exact entanglement....", but what is calculated is the second Renyi mutual information. I did not find a statement of how this Renyi mutual information is related to the entanglement. This should be clarified. If it is just the authors calling this mutual information "exact entanglement", that does not seem appropriate to me.
In the abstract, they suggest that the volume law "might hold for more general mesoscopic systems." This seems likely if one restricts to noninteracting systems. But Ref. 13 seems to find that interactions reduce the mutual information down to area law. So here in the abstract, the word "noninteracting" should be added, unless the authors do mean also interacting systems and can support that position and point out how Ref. 13 was wrong about this.
It seems that the results are for the long-time NESS, and not for earlier times. I did not find this stated explicitly; it should be made more clear.
I suspect that nonGuassian initial states will approach Gaussian at late times, which is what justifies restricting attention to Gaussian states; again this should be stated more clearly. Or maybe the authors intended to restrict to only Gaussian initial states? If so, please clarify.
Just above Eq. 3, the interval considered is said to be within [0,1], but up to that point it is a lattice model with only integer positions up to N. Probably something like x=j/N is meant and the interval is being given in terms of x, the scaled length (?). This should be stated explicitly.
The second Renyi mutual information does not have all the properties of a proper mutual information. Can the authors make any definite statements about the true (von Neumann) mutual information? Even if not, the von Neumann should be at least mentioned, if only to state that no results for it will be presented.
In the conclusion section, it says "...mesoscopic regime, with a coherence length smaller than the....", which seems to contradict the definition stated in footnote 3 of the introduction. Probably this is a "typo" in the conclusions; if so it should be fixed (or clarified).