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Symmetry-resolved dynamical purification in synthetic quantum matter
by Vittorio Vitale, Andreas Elben, Richard Kueng, Antoine Neven, Jose Carrasco, Barbara Kraus, Peter Zoller, Pasquale Calabrese, Benoit Vermersch, Marcello Dalmonte
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Submission summary
| Authors (as registered SciPost users): | Marcello Dalmonte · Vittorio Vitale |
| Submission information | |
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| Preprint Link: | https://arxiv.org/abs/2101.07814v2 (pdf) |
| Date submitted: | 2021-12-10 10:17 |
| Submitted by: | Vitale, Vittorio |
| Submitted to: | SciPost Physics |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
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Abstract
When a quantum system initialized in a product state is subjected to either coherent or incoherent dynamics, the entropy of any of its connected partitions generically increases as a function of time, signalling the inevitable spreading of (quantum) information throughout the system. Here, we show that, in the presence of continuous symmetries and under ubiquitous experimental conditions, symmetry-resolved information spreading is inhibited due to the competition of coherent and incoherent dynamics: in given quantum number sectors, entropy decreases as a function of time, signalling dynamical purification. Such dynamical purification bridges between two distinct short and intermediate time regimes, characterized by a log-volume and log-area entropy law, respectively. It is generic to symmetric quantum evolution, and as such occurs for different partition geometry and topology, and classes of (local) Liouville dynamics. We then develop a protocol to measure symmetry-resolved entropies and negativities in synthetic quantum systems based on the random unitary toolbox, and demonstrate the generality of dynamical purification using experimental data from trapped ion experiments [Brydges et al., Science 364, 260 (2019)]. Our work shows that symmetry plays a key role as a magnifying glass to characterize many-body dynamics in open quantum systems, and, in particular, in noisy-intermediate scale quantum devices.
Current status:
Reports on this Submission
Report #4 by Anonymous (Referee 2) on 2022-2-4 (Invited Report)
- Cite as: Anonymous, Report on arXiv:2101.07814v2, delivered 2022-02-04, doi: 10.21468/SciPost.Report.4315
Report
This a very good paper on a timely and hot subject. The authors show that contrary to intuitions, in the presence of continuous symmetries and under ubiquitous experimental conditions, symmetry-resolved information spreading is inhibited due to the competition of coherent and incoherent dynamics: in given quantum number sectors, entropy decreases as a function of time, signalling what they call "dynamical purification". They then propose to use their random unitaries toolbox and apply to experimental date on trapped ions. This is a top class research and paper shoudl be published . I have one minor remark: to make a reference list more complete i would suggest to cite tiogehter with [90], the even newer review: 23. Monika Aidelsburger, Luca Barbiero, Alejandro Bermudez, Titas Chanda, Alexandre Dauphin, Daniel González-Cuadra, Przemysław R. Grzybowski, Simon Hands, Fred Jendrzejewski, Johannes Jünemann, Gediminas Juzeliunas, Valentin Kasper, Angelo Piga, Shi-Ju Ran, Matteo Rizzi, Gérman Sierra, Luca Tagliacozzo, Emanuele Tirrito, Torsten V. Zache, Jakub Zakrzewski, Erez Zohar, and Maciej Lewenstein, Cold atoms meet lattice gauge theory, Phil. Trans. R. Soc. A 380, 20210064 (2021), http://doi.org/10.1098/rsta.2021.0064 (2022), arXiv:2106.03063.
Report #1 by Anonymous (Referee 3) on 2022-1-17 (Invited Report)
- Cite as: Anonymous, Report on arXiv:2101.07814v2, delivered 2022-01-17, doi: 10.21468/SciPost.Report.4063
Strengths
1) Discovery of a qualitatively new event, namely that the competition between coherent and dissipative dynamics in an open many body system may lead at intermediate times to increased purity and entanglement in certain symmetry sectors although overall state becomes less pure with time.
2) Extensive set of examples demonstrating the behavior, including both numerics on a variety of realistic system and new analysis of previous experimental data.
Weaknesses
1) The restrictions on the dissipation to be compatible with the symmetry are not clarified.
2) The scaling of many of the features with the subsystem size and other parameters is not given explicitly.
3) The probability of being in a symmetry sector undergoing dynamical purification is not presented in most examples.
Report
The study of entanglement in many body systems in general, and its interplay with symmetries and conservation laws in particular, has received much attention recently, especially since the beginning of the NISQ era. To the best of my knowledge, this work presents a qualitatively new effect in this field, namely that in a many body system evolving under a combination of coherent and dissipative dynamics, while the overall purity and entanglement of a subsystem tend to decrease with time, their values in particular symmetry sectors may actually rise at intermediate times. The Authors give simple perturbative arguments for this behavior, then demonstrate it for a wide spectrum of examples, including numerics on a variety of experimentally-realizable systems, as well analysis of data from a previous experiment. The manuscript is also clearly-written. I therefore believe this work warrants publication in SciPost Phys. However, some points should be addressed first:
1) As explained by the Authors, for the effect to take place the coherent dynamics should obey the symmetry while the dissipation should break it. However, it should be noted that the dissipation should still preserve the block structure of the density matrix of a subsystem (although it should cause transitions between blocks); such a situation is known as ``weak symmetry’’, see, e.g.,
B. Buča and T. Prosen, A note on symmetry reductions of the
Lindblad equation: Transport in constrained open spin chains,
New J. Phys. 14, 073007 (2012);
V. V. Albert and L. Jiang, Symmetries and conserved quantities
in Lindblad master equations, Phys. Rev. A 89, 022118 (2014);
and later works. The Authors should discuss the notion of weak symmetry and its role in their effect.
2) Most of the discussion concentrates on the symmetry sector which ``neighbors’’ the one which is initially populated, though it is mentioned and exemplified that similar effects may occur in other sectors. Could the Authors estimate the number of sectors in which dynamical purification may occur and its scaling with the parameters and subsystem size?
It would also be useful to have explicit expressions for the scaling of the time of maximal purity, the corresponding maximal value, and the width of the peak with the various parameters and subsystem size – some sentences in Sec. 3.1 seem to indicate the Authors are somewhat reluctant to give such expressions, but I do believe that if possible, it would be useful for future readers.
3) The observability of the discussed effect depends on the probability of experimentally finding the system in a sector experiencing dynamical purification (which amounts to post-selection). However, this information is presented only for the experimental data in Sec. 6. It should be added for the analytical estimate in Sec. 3 all the numerical examples in Sec. 4.
Requested changes
1) Adding a discussion of weak symmetries and their role in the results
2) If possible, adding a discussion of the scaling of various features with the subsystem size and other parameters, especially the number of sectors experiencing dynamical purification.
3) Adding the probabilities of finding the system in a symmetry sector undergoing dynamical purification.
Author: Vittorio Vitale on 2022-02-14 [id 2196]
(in reply to Report 1 on 2022-01-17)
We thank the Referee for their careful reading of the manuscript, and for their comments, that we address below point by point.
Point 1:
Indeed, as pointed out by the Referee, we are considering a system where the dissipator shall satisfy weak symmetry: for instance, in our Bose-Hubbard model, the effect of a jump operator of the form $b^\dagger +b $ would not be captured by our theory.
We have clarified this aspect in two parts of the manuscript:
a) at the beginning of Sec. 3.1, where the general conditions for the theory are presented (we have also included the references suggested by the Reviewer), where we have also emphasized that this choice reflects typical experimental conditions;
b) after Eq. 17, in a footnote we introduced.
In fact, dynamical purification can also occur even if weak symmetry is broken, as shown in our experimental example. Of course, in that case, the notion of entropy shall be handled with care, as the density matrix is not in block-diagonal form: still, symmetry resolved quantities have specific operational meaning even in that context, as we have elaborated upon in Npj Quantum Inf. 7, 152 (2021) (in particular, the offer meaningful bounds to information witnesses).
Point 2:
The Referee raises a very important point, partly connected to their third comment as well. Let us split the question into two parts - different sectors, and scaling of maximal purity.
A) In general, the principle that governs dynamical purification also applies to sectors that are not neighbors of the initially occupied one: first, dissipation scrambles information in the sector, and then, coherent dynamics partly re-order the latter. However, a clean mathematical understanding of this phenomenon is a non-trivial task.
The reason is the following: while NN sectors can be rigorously treated with second order perturbation theory, going beyond this would require at least third order, or even higher orders in 1D (since tunneling is strongly limited). This seems, in principle, technically feasible: however, empirically, we found such effects to be quite small (one example is depicted in Fig. 7), so we decided not to focus on those, as they will likely have limited experimental applicability.
B) Maximal purity: this is an extremely interesting question, that we had tried to address within out framework. Unfortunately, it turns out that the 'peak' region is something that, on its own, cannot be immediately captured by perturbation theory: it is exactly the timescale where states that belong to generic subspaces $E_2(-1), E_3(-1)$ become important, because of both (i) dynamics inside the partition and (ii) multiple dissipative events. Both of these effects are well beyond our theory, so we are unable to capture neither the position nor the maximum of the peak with precision. We are only able to determine the scaling functions (see Eq. 23). Empirically, we have observed that the position of the maximum seems relatively unaffected by noise and even interactions in both XY, Bose-Hubbard, and U(1) lattice gauge theories (Fig. 4, 5, 6), so this may suggest some generic scaling, but as it stands, this shall be understood as a numerical observation.
Point 3:
We have added the population dynamics also for some sample cases of our simulations (most models follow the same pattern) with a new appendix including 3 figures, spanning several parameter regimes. We have also commented that, in the regime of dynamical purification, the probability of being in a sector that is neighbor to the starting one is expected to grow linearly with time, and then enter a quadratic regime, similarly to the prefactors $A(t)$.
Report #2 by Anonymous (Referee 5) on 2022-1-3 (Invited Report)
- Cite as: Anonymous, Report on arXiv:2101.07814v2, delivered 2022-01-03, doi: 10.21468/SciPost.Report.4133
Strengths
1-The paper introduce a procedure to analyse the quantum information content
in distinct sectors of quantum systems with U(1) symmetry.
2- The theoretical calculations are done in several distinct systems, indicating the generality of their results.
3- They uses experimental results already known in the literature to corroborate their theoretical discovers.
4-There is a potential that this paper will motivate further experimental set up in the arena of quantum information.
Report
The paper certainly is in a good level and should be published
Requested changes
Revise for some trivial and minor typos.
Author: Vittorio Vitale on 2022-02-14 [id 2195]
(in reply to Report 2 on 2022-01-03)
We thank the Referee for their careful reading of the manuscript, and for the feedback.
We have made an extra effort to correct a few remaining typos.
Author: Vittorio Vitale on 2022-02-14 [id 2197]
(in reply to Report 4 on 2022-02-04)We thank the Referee for their careful reading of the manuscript, and for the positive assessment. We have included the paper suggested to our bibliography together with the previously cited review article.