SciPost Submission Page
Estimating time in quantum chaotic systems and black holes
by Haifeng Tang, Shreya Vardhan, and Jinzhao Wang
This is not the latest submitted version.
Submission summary
| Authors (as registered SciPost users): | Shreya Vardhan |
| Submission information | |
|---|---|
| Preprint Link: | scipost_202505_00015v1 (pdf) |
| Date submitted: | May 8, 2025, 8:30 p.m. |
| Submitted by: | Shreya Vardhan |
| Submitted to: | SciPost Physics |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
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| Approaches: | Theoretical, Computational |
Abstract
We characterize new universal features of the dynamics of chaotic quantum many-body systems, by considering a hypothetical task of ``time estimation." Most macroscopic observables in a chaotic system equilibrate to nearly constant late-time values. Intuitively, it should become increasingly difficult to estimate the precise value of time by making measurements on the state. We use a quantity called the Fisher information from quantum metrology to quantify the minimum uncertainty in estimating time. Due to unitarity, the uncertainty in the time estimate does not grow with time if we have access to optimal measurements on the full system. Restricting the measurements to act on a small subsystem or to have low computational complexity leads to results expected from equilibration, where the time uncertainty becomes large at late times. With optimal measurements on a subsystem larger than half of the system, we regain the ability to estimate the time very precisely, even at late times. \\ Hawking's calculation for the reduced density matrix of the black hole radiation in semiclassical gravity contradicts our general predictions for unitary quantum chaotic systems. Hawking's state always has a large uncertainty for attempts to estimate the time using the radiation, whereas our general results imply that the uncertainty should become small after the Page time. This gives a new version of the black hole information loss paradox in terms of the time estimation task. By restricting to simple measurements on the radiation, the time uncertainty becomes large. This indicates from a new perspective that the observations of computationally bounded agents are consistent with the semiclassical effective description of gravity.
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
Report #2 by Anonymous (Referee 2) on 2025-7-25 (Invited Report)
- Cite as: Anonymous, Report on arXiv:scipost_202505_00015v1, delivered 2025-07-25, doi: 10.21468/SciPost.Report.11632
Strengths
2. The dependence of the time evolution and late-time scaling laws of the QFI on subsystem size and on whether the dynamics are chaotic or integrable is clearly exposed.
3. The combination of spin chains, Lindbladians, random Schmidt-basis states, and Brownian GUE toy models provides rich and convincing interpretations of the observed phenomena.
4. The contrast drawn between QFI and CFI, and the experiment on estimating time are timely and well motivated.
Weaknesses
2. The black-hole section is under-specified and partially unjustified.
a. The gravitational setup is not fully stated: the gravity-plus-matter theory, spacetime asymptotics, the preparation and the age of the black hole should be specified.
b. Eq. (21) appears to be an expectation inferred from Eq. (20), yet it is used as the key step in deriving Eq. (26).
c. It is unclear why the volume of the photon gas should be proportional to the evaporation time in Eq. (24). Instead, the photon energy in Eq. (22) should be tied to the energy lost by the black hole.
d. Black hole evaporation is a nonequilibrium process. The manuscript seems to model the evaporation during a short window δt ≪ δt_evap as nearly equilibrium. How can this be justified or checked?
Report
The work is conceptually strong and potentially impactful for both quantum many-body physics and black-hole information. I find it suitable for publication in SciPost, subject to the requested changes below.
The authors are also asked to address two specific questions:
1. In the analytical derivations, the boundary contribution is assumed negligible compared to the bulk, |∂S| ≪ n_S. Does this imply that the results do not extend to all-to-all Hamiltonians, such as the SYK model?
2. I agree that entanglement entropy is insensitive to rotations of the support of the reduced density matrix on a subsystem. If I understand correctly, these rotations only set the overall energy-density-squared scale of the QFI, while the universal terms depend solely on n_A and n_{\bar A} and are thus captured by the entanglement entropy. Consequently, once the dynamics generate a 2-design at late times, the precise form of the Hamiltonian becomes unimportant. Is this why random pure states and the Brownian GUE toy model—both independent of microscopic Hamiltonian—work so well for computing the QFI?
Requested changes
1. In connection with Weakness 1, clearly distinguish results obtained numerically from those derived analytically, and highlight all assumptions used in the derivations in the main text.
2. In connection with Weakness 2, spell out the gravitational setup in detail and provide further justification for Eqs. (21)–(25).
3. Correct the typo “chaotiic” on line 372 to “chaotic”.
Recommendation
Ask for minor revision
Report #1 by Anonymous (Referee 1) on 2025-7-17 (Invited Report)
- Cite as: Anonymous, Report on arXiv:scipost_202505_00015v1, delivered 2025-07-17, doi: 10.21468/SciPost.Report.11585
Strengths
- Introduce a new metric to probe quantum chaos and thermalization dynamics, based on estimating time from local / low-complexity measurements.
- The paper is largely clear and well-written.
Weaknesses
- Many of the claims are based on numerics only, and it is often unclear what results can be derived analytically for some specific models of dynamics, and what results were just inferred from numerics.
- Results for integrable systems are limited to non-interacting (free fermions) systems, and aren't representative of generic integrable systems.
- Sections 4,5,6 feel a bit disconnected from the rest of the paper, and could be either appendices or be integrated better with the rest of the narrative.
Report
I think this is a very good paper, that can be published in Scipost Physics. I have a few suggestions listed below that I believe could improve the paper before publication.
Requested changes
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While the paper is well-written overall, I felt that sections 4-6 were a bit disconnected from the main narrative. Perhaps some of the content of these sections could be integrated in the rest of the text in a better way. For example, I think section 5 (which provides a practical implementation of the "estimating time" idea) could come a lot earlier to illustrate the authors' main idea.
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The authors argue that their quantity is useful to distinguish integrable and chaotic behavior. However, that claim is based on taking a non-interacting free fermion chain as a model of "integrability". This can be very misleading, as free fermion systems are very much not representative of interacting integrable systems, which can have very different behavior in quantities like OTOCs. I would ask the authors to either change the terminology from "integrable" to "free fermions", or add numerics on interacting integrable systems.
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Are all universal results on the scaling of the subsystem QFI inferred from numerical observations? Some of the results follow from the Brownian dynamics model described in an appendix for example. It might be useful to clarify which result (if any) can be established analytically (even for some toy models), and which results are purely based on small system numerics.
Recommendation
Ask for minor revision
Author: Shreya Vardhan on 2025-08-18 [id 5737]
(in reply to Report 2 on 2025-07-25)In response to the specific questions in this report (see the resubmission for a full list of changes in response to the other comments):
It is true that the result in equation (22) does not apply to the SYK model. See the comment added in footnote 7 on page 16 of the resubmitted version about how one can derive the analogous result for SYK.
It is not true that the final result, say in equation (22), can be expressed in terms of entanglement entropies of subsystems. Note that the entanglement entropies for random pure states are always proportional to $\text{min}(n_A, n_{\bar A})$, and never provide information about the size of the larger of the two subsystems. In contrast, the second line of (22) is proportional to the size of the larger subsystem. This is a concrete way of seeing that the QFI has access to information about the system beyond that captured by the entanglement entropy.
"Consequently, once the dynamics generate a 2-design at late times, the precise form of the Hamiltonian becomes unimportant."
It is not immediately clear whether the relevant time scale for saturation of the QFI is the one where the time-evolution unitary forms a 2-design: note that the $U$ and $V$ that appear in equation (21), whose 2-design properties used, are not not quite the same as the time-evolution unitary. Instead, these are the unitaries that relate some simple reference vectors in $A$ and $\bar A$ to the Schmidt vectors of the time-evolved state in $A$ and $\bar A$.
"Is this why random pure states and the Brownian GUE toy model—both independent of microscopic Hamiltonian—work so well for computing the QFI?"
We expect that the reason why random pure states work well for estimating the late-time QFI (and in particular, match well with the spin chain numerics) is that in the spin chain system, we have chosen an ensemble of initial random product states, which are expected to equilibrate to infinite temperature and resemble random pure states at late times.