Nonlocality of Deep Thermalization

Referee report 1 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 thermalisation usually means Lieb-Robinson bounds. Do the authors claim that this is violated in some sense? Or is the claim simply that deep thermalisation 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.

Response: We thanks the reviewer for their comments. The reviewer is correct in pointing out that Lieb-Robinson (LR) bounds are governed by local interaction terms in a many-body system. LR bounds govern the rate at which entanglement or entanglement entropy can grow in the many-body system and as a result also determine the rate of regular thermalisation. This rate is independent of the boundary conditions of the infinite 1D chain considered in our work, indeed we find that regular thermalisation occurs exactly in time t_1 = ceil(N_A/2) for both PBC and OBC. However, the rates of deep thermalisation (as probed by moments k=2 and higher) are clearly found to be different (rate of deep thermalisation being twice as fast for PBC than for OBC) which highlights the fact that deep thermalisation is not constrained by the same information propagation bounds (LR bounds) as regular thermalisation and probes novel physics that goes beyond regular thermalisation. Below we address the issues raised by the referee.

1 - Explain advance beyond [11,12] that meets SciPost Physics acceptance criteria

Response: Firstly in the setup considered here, for the first time we concretely show that the time scales for regular and deep thermalisation are different by studying the emergence of deep thermalisation for a projected ensemble within the bulk of a many-body system. Secondly, both of these references do not consider the dependence of Deep thermalisation on boundary conditions. Note that studying this dependence on boundary conditions is absolutely crucial since it brings out the difference in rates of deep thermalisation in the two boundary conditions considered here. Further, this difference in rates is attributed to the non-locality of deep thermalisation which is the main essence of our work. With the above clarification and the responses below, we believe that the essential criteria for publication that is met by our work is: “Open a new pathway in an existing or a new research direction, with clear potential for multi-pronged follow-up work”.

2- Explain in more detail what is meant by non-locality, how it is different from the standard notion in thermalization, and why the result is interesting/surprising.

Response: Following from the first response it is important to note that the rate of regular thermalization (k=1) is the same for both PBC and OBC so probing the rate of local observables reaching their thermal values is not enough to bring out this interesting aspect of deep thermalisation. We have emphasised this point in the Discussion section as well. The difference in rates of deep thermalisation for PBC vs OBC only comes about for k=2 and higher moments which shows that deep thermalisation is intrinsically different from regular thermalization. Having established the distinction from regular thermalisation, due to locality constraints LR bounds govern the speed at which correlations grow during chaotic dynamics; despite that the boundary effects (of an infinite chain) show up in the rates of deep thermalisation for PBC vs OBC; this is what highlights the non-locality of deep thermalization.

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

Response: We have removed jargon wherever suitable.

Referee report 2 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. Response: We thank the reviewer for their appraisal and comments. We address the points below

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. |+> vs |0> ?

Response: Note that here we make use of the result that in the thermodynamic limit of N_B1 and N_B2 -> \infinity, the unitary ops U_z1 and U_z2 can be replaced by independent instances of Haar random unitary ops U and U’. Then in the dual picture these Haar random unitary ops act on a state of t-qubits to produce resultant states |U> and |U’> for OBC as shown in Fig. 2c. Since the resultant ensemble of states over many instances of U and U’ is independent of the initial state, we pick states |+>^{\oprod t} and |0>^{\oprod t} from two mutually unbiased bases (X and Z) to highlight the fact that there is nothing special about choosing the initial states on the two ends and the result is independent of it.

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?

Response: The reviewer is correct in pointing out that the number of off-diagonal terms are generally much more than diagonal terms. However, for any finite moment ‘k’ the number of such off-diagonal terms are countably many and each of them is smaller than diagonal terms by at least a factor 2^{2t} for PBC and 2^{t} for OBC. Crucially, the number of such terms does not change over time. So, even if the norm of the off-diagonal terms is close to 1 (after normalising \rho^{(k,n)}) for small times, this norm shrinks over time exponentially fast and as a result becomes infinitesimally small for large times.

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 ceil(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?

Response: We thank the reviewer for the interesting remark. Since regular thermalization is governed by local interactions - a region of interest buried deep in the bulk has 2 boundaries in contact with the bath which could influence the rate of regular thermalization via faster entanglement growth which explains the shorter time required for regular thermalization. However, in our study we consider a region buried in the bulk and compare rates of deep thermalization for periodic and open boundary conditions. For an infinite 1D chain of spins, the boundaries are far outside the effective light cone despite which the boundary conditions influence the rate of deep thermalization (but not of regular thermalization). Therefore, we believe that deep thermalization probes novel physics that goes beyond regular thermalization.

4- Could the authors comment on the effect that more involved boundary conditions, e.g. twisted boundary conditions, would have in this setup?

Response: We note that for twisted boundary conditions (TBC), the Ising interaction strength only for the sites at extreme left and right ends of the chain is modified from J to J*exp(i\phi). In the tensor network diagram of Fig. 2 this contributes a multiplicative factor and a phase factor at the left and right ends of the circuit. The multiplicative factor gets adjusted during normalisation and the extra phase factor on the boundary can be absorbed into the Haar unitary operators after taking the thermodynamic limit of bath. Thus we expect the rates of deep thermalization for PBC and TBC to be same.

As per the suggestions of the referees we have made the following changes:
1. Eliminate jargon in places where terminology could be simplified
2. Formatting changes for proper alignment of certain figures