Nonlocality of Deep Thermalization

1- Analytical results on deep thermalization rates in a system with both open and closed boundary conditions.
2- Well written and timely.

1- Nonstandard use of "locality" and "topology".
2- Some technical aspects could be clarified.

In this work the authors study the phenomenon of 'deep thermalization': in the dynamics of many-body quantum systems, after projective measurements on a bath the corresponding distribution of states on a local subsystem is expected to reach a universal and maximally entropic distribution (i.e. the projected ensemble approaches the Haar ensemble) after a sufficiently long time. Specifically, the authors consider a subsystem located in the bulk of a one-dimensional lattice system with dynamics governed by the kicked Ising model. This model was one of the first for which deep thermalization was established, and where previous studies focused on subsystems located at the edge of the lattice, this focus on the bulk allows the authors to probe the effect of boundary conditions. They find that the rate at which deep thermalization occurs depends crucially on the choice of boundary conditions ($t$ vs. $2t$ for open and periodic boundary conditions respectively).

The authors make use of the fact that the system under study satisfies a notion of 'dual-unitarity', which combined with a replica trick allows for analytical calculations of the k-th moment operators characterizing the projected ensemble. The analytical results depend on the validity of the (commonly used) replica trick and are supported by numerical simulations.

The main contributions of this work are twofold. First, analytical results are presented showing deep thermalization in the considered systems, where the projected ensemble approaches the Haar ensemble exponentially fast. Furthermore, deep thermalization occurs at a nontrivial time scale different from that of regular thermalization, unlike in previous studies of this model where all time scales collapse and trivialize. Second, it is shown that the rate at which this occurs depends crucially on the boundary conditions. This highlights that deep thermalization is not a "local" phenomenon — unlike regular thermalization, where the relevant time scale does not depend on the choice of boundary conditions.

I have no doubt that these results are valid and interesting, and I am happy to recommend this paper for publication (with some very minor questions). However, I agree with Anonymous Report 1 that a better case could be made as to why this paper should be published in SciPost Physics rather than SciPost Physics Core. While the results are interesting, the authors could make a stronger case as to how this paper meets the SciPost Physics acceptance criteria. Otherwise all acceptance criteria for publication are clearly met.

I have some minor comments/questions, which are mainly meant for clarification and do not impact my judgement of this paper.

1- On page 10, below Eq. (16), why are the closed boundary conditions different to the left and the right, i.e. $|+\rangle$ vs. $|0\rangle$?
2- In deriving the convergence in time to the Haar ensemble, the authors show that individual off-diagonal contributions are exponentially suppressed compared to the diagonal contributions. However, it also seems that the number of off-diagonal contributions can be exponentially larger than the number of diagonal contributions. Could the authors comment on this?
3- In the case of open boundary conditions, the time for exact thermalization (rather than deep thermalization) in the considered model depends on whether the considered subsystem is located in the bulk or at the edge of the lattice. In the former case the timescale is $N_A/2$, as in the current work, whereas in the latter case the thermalization time is twice that — so again a time $t$ vs. $2t$. Is there some connection with the current results?
4- Could the authors comment on the effect that more involved boundary conditions, e.g. twisted boundary conditions, would have in this setup?

1- Very pedagogical introduction to a topical subject

1- Despite being generally pedagogical, jargon obfuscates the statements at certain points.
2- Unclear whether contribution of manuscript goes sufficiently beyond [11,12] to justify the SciPost Physics acceptance criteria.
3- "Locality" seems to be used in non-standard way.

In this paper the authors study non-locality in deep thermalization. Deep thermalization is a topical subject. The authors provide a nice introduction to the subject and use techniques based on dual unitarity to give exact results for deep thermalization for two different boundary conditions. The find difference between the rate of deep thermalization and conclude that this is a manifestation of non-locality of deep thermalization.

I have enjoyed reading this paper. It is very nicely written for the most part. However, I still have many issues with the paper meeting the very stringent SciPost Physics acceptance criteria.

It is unclear to me in what sense is their effect non-local. The term locality in connection to dynamics and thermalization usually means Lieb-Robinson bounds. Do the authors claim that this is violated in some sense? Or is the claim simply that deep thermalization inherently probes non-local information? I agree with the latter claim and I do not find it very surprising. The authors do touch on this in Discussion, but the conclusion was not very clear to me.

1 - Explain advance beyond [11,12] that meets SciPost Physics acceptance criteria
2- Explain in more detail what is meant by non-locality, how it is different from the standard notion in thermalization, and why is the result interesting/surprising.

Minor:
3-Eliminate jargon where possible, e.g. what is the definition of "fiducial state".