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Generalized Lindblad Master Equation for Measurement-Induced Phase Transition
by Yi-Neng Zhou
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Submission summary
| Authors (as registered SciPost users): | Yi-Neng Zhou |
| Submission information | |
|---|---|
| Preprint Link: | https://arxiv.org/abs/2204.09049v3 (pdf) |
| Date accepted: | 2023-01-23 |
| Date submitted: | 2023-01-03 08:06 |
| Submitted by: | Zhou, Yi-Neng |
| Submitted to: | SciPost Physics Core |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
Abstract
The measurement-induced phase transition (MIPT) occurs when the system is evolving under unitary evolution together with local measurements followed by post-selection. We propose a generalized version of the Lindblad master equation as a continuous equation, to describe the dynamics of second R\'enyi entropy in the MIPT. This generalized Lindblad equation explicitly takes into account the post-selection in the MIPT, which is realized as the Einstein-Podolsky-Rosen (EPR) state projection in the equation. Also, this generalized Lindblad equation preserves the Hermitian, unit trace, and positive definiteness of the density matrix. We further use the hard-core Bose-Hubbard model as a concrete example to numerically confirm that our generalized Lindblad equation is applicable to describing the MIPT.
Author comments upon resubmission
Thank you very much for handling and reviewing our manuscript. We have revised our manuscript based on the reports of the referee. Here, we resubmit our paper entitled “Generalized Lindblad Master Equation for Measurement-Induced Phase Transition” for your consideration.
Our paper has been reviewed by one Referee. The referee finds our work interesting and scientifically sound, and the topic is extremely timely.
We also received invaluable suggestions from the referee. In the resubmitted manuscript, we have carefully revised our paper based on the report. In particular, we add a new discussion in section 3 of the main text and Sec. I of the supplementary about the difference between Eq. (6) and the “weak measurement” scenario which generally does not assume the projection on a complete basis. We add more numerical results using the quantum trajectory methods in section E of the supplementary material and the discussion about the results and the possible entanglement phase transition. We also rewrote some sentences in the Sec.7 to be more prudent about how our numerical results are supportive of a MIPT.
We, therefore, resubmit the paper to SciPost and hope this revision can receive your and Referee’s favorable consideration.
Best Regards,
Yi-Neng Zhou
List of changes
1. I have added the discussion in section 3 of the main text and section I of the supplementary about the difference between Eq. (6) and the “weak measurement” scenario which generally does not assume the projection on a complete basis.
The discussion about the difference between Eq. (6) and the “weak measurement” scenario is mainly the following:
In deriving the Lindblad-like equation in the section.2, we assume that the measurement is performed on a complete basis, thus the anti-commutator part of the Lindblad master equation is trivial and amounts to an identity matrix, Thus, Eq. (6) is a special case of the general Lindblad equation. Also, the probabilistic measurement process in Eq. (4) leads to the anti-commutator part of the Lindblad master equation amounts to an identity matrix.
If we do not assume this probabilistic measurement process, the system under unitary evolution together with the measurement process can be described in other ways, and we do not need to assume the completeness condition the Eq. (5).
2.I have cited the list of references that the referee has recommended as the ref [12],[31],[32].
3. I have rewritten some sentences in the Sec.7 to be more prudent about how our numerical results are supportive of a MIPT. Thanks to the referee for reminding me of this.
4. In this revision, I have added some numerical results and a discussion about the change of the longtime saturation value of entanglement entropy as the total system size increases using the quantum trajectory method. Also, the details of the quantum trajectory method we have used here and more numerical results are included in section E of the supplementary material.
5. Thanks to the referee for reminding me of the typos, and I have fixed all the typos about MIPT.
Published as SciPost Phys. Core 6, 023 (2023)