Equilibration of Non-Markovian Quantum Processes in Finite Time Intervals

The manuscript by Dowling et al. reports results on the equilibration of quantum processes in finite time intervals. The work addresses a rather natural extension of a paper by the same authors (Ref. [14]) in which equilibration in the infinite-time-interval limit was discussed. While the question addressed in the present paper is a rather straightforward generalisation, it requires different methods and strategies to make progress on the technical side. I consider the paper a worthwhile study that should be published in some form. I do have a number of comments and requests that I would like to see addressed. While most of these comments in the below list concern minor issues regarding the presentation or notation, the first item on the list, while possibly easy to resolve, is a bit more substantial.

Once these issues are resolved, the paper certainly merits publication in some journal. Whether a paper like this, which extends an earlier result in a natural way, meets the bar for publication in SciPost Physics, I find much harder to answer. SciPost's requirements of a "groundbreaking [...] discovery" or "breakthrough on a previously-identified and long-standing research stumbling block" is not quite what I see in the paper. But a solid piece of research that makes progress in an interesting direction.

(a) Section IV, Eq. (51): According to the text below the equation, the first inequality in that equation is obtained by using "that $\mathcal{E}_{\mathcal{M}}$ is a minimum over all processes in the restricted set $\mathcal{K}$". This implies that each of the two minimisation functionals in the second line of (51) can be upperbounded by $\mathcal{D}_{\mathcal{M}}$. But the second minimization comes with a minus sign, and I don't see why the DIFFERENCE of the two minimisations should be upperbounded by the difference given in the third line of (51). The fact that, by assumption, $\mathcal{E}_{\mathcal{M}}(\Upsilon) \geq \mathcal{E}_{\mathcal{M}}(\Omega)$ doesn't help either. What am I missing?

An additional comment: While in the second line of (51) a minimum is taken over $\Lambda_\Omega$, the third line now suddenly seems to depend on $\Lambda_\Omega$. If this is the case, I'd assume that what is meant here is that the third line holds for arbitrary $\Lambda_\Omega \in \mathcal{K}$. Is this correct? Then this must be stated explicitly!

(b) The title sounds as if the results were restricted to non-Markovian processes, which, to my understanding, is not the case. "Equilibration of Quantum Processes in Finite Time Intervals" would then be a more accurate description of the content of the paper.

(c) While most of the paper is well written and clear, I found the abstract quite confusing, and pretty much impossible to understand before having read the paper. The first sentence makes a statement that is wrong in its absoluteness. Only the second sentence resolves this. Start the abstract by saying "Under suitable conditions, quantum processes...". Then: "Sufficient conditions for this result to hold are multitime observables that are coarse grained...". The next sentences starts with "This dictates...", but I don't find it obvious what "this" here is referring to. The conditions mentioned in the previous sentence? Or the result mentioned in the first sentence? Same in the fourth sentence: "We show that this leads to...". Which "this"? There is quite some room to improve the clarity and readability of the abstract.

(d) Third paragraph of the Introduction: "This IS a foundational question..."

(e) Same paragraph: What does "significantly interacting energy eigenstates" mean? States aren't interacting, I'd say. And if one writes a Hamiltonian in its energy eigenbasis, then there are no interacting terms in that basis. Please explain!

(f) Eq. (4): The notation $\mathbb{1}_2$ is so commonly used for the identity operator on $\mathbb{C}^2$ that I was pretty confused by the equation. Please introduce the notation here.

(g) Eq. (5): Same here, please briefly explain $^{T_1}$ notation. Sure, one can figure it out from the context, but a reader-friendly comment would be appreciated.

(h) Below Eq. (13): Does "trace properties" here mean "trace preserving properties"? If so, the latter would be clearer.

(i) Right thereafter: "...the latter is crucial for ensuring the causality of the process." I am surprised that trace properties ensure causality. Is there an argument that could be given as to why this is the case?

(j) Eq. (14): The notation $_{\Upsilon-\Omega}$ needs some guesswork as to what is meant. Please define!

(k) Below Eq. (17): "...as the average indistinguishability...". Please specify what average is meant here. (Time average, I suppose!?).

(l) First sentence of Sec. III: dotted --> dashed

(m) Eq. (23): Define $^\vec{T}$ notation. I don't think $\vec{T}$ had been used before.

(n) Below Eq. (26): "Here, $\mathcal{S}_{\mathcal{M}}$ is the total combined number of outcomes for all instruments..." I find this unclear, especially the "total combined". Please give a mathematical expression for $\mathcal{S}_{\mathcal{M}}$, I'd hope this makes it clearer.

1. Rigorous analytical results on the timescale at which “multi-time processes” are effectively in equilibrium.

2. Very general set-up.

3. Well written paper, with mostly appropriate referencing.

1. The results follow from previous proofs in the literature, and there are no new techniques for analyzing quantum dynamics.

2. Setting and results not completely well motivated from the conceptual point of view.

In this work, the authors study the notion of multi-time processes, which can be understood as a series of quantum dynamics between which an undetermined measurement or intervention is placed. Here, they consider the case in which the dynamics are unitary and generated by a time independent Hamiltonian.

These are of interest because in the context of many body dynamics, it is expected that many interesting models reach an effective equilibrium state, in which they stay for very long times. The question typically analyzed in the literature is: let a state evolve for time t, at which we measure a given observable. Does the expectation value of that observable correspond to the equilibrium state? Moreover, when is that effective equilibrium reached?

This is a timely question and a fair amount of work along different lines has been devoted to it. This includes a series of analytical works including references [10-13,15,37-38,42-47]. The present paper changes the question slightly, by generalizing it to scenarios where multiple measurements at different time intervals can happen, as described by the formalism of process tensors. The paper is well written and structured, and the key results used are referenced accordingly. It perhaps misses some discussion or citations about the wider literature on quantum dynamics and thermalization beyond the aforementioned papers (which are on a specific line of work within it).

In a previous paper [14], the authors studied the same setting, and prove a bound on the fluctuations away from equilibrium that holds for all times, following the proof idea of [11-13]. Here, instead, they consider the fluctuations for finite times. They show that an extension of [42] holds in this case. The proof follows [42] straightforwardly, with the complication that now there are “k” times to consider, and the number of terms increases accordingly. The complication is dealt with through various groupings of terms and chains of inequalities. The main idea is to consider the cases of small “k” first, and then explain how they generalize.

The technical part of the paper is largely about finding the way to bound the quantities considered to the point that the results [42] can be applied. A perhaps interesting point is that this more general result allows them to consider a diamond norm distance and other nontrivial figures of merit, as opposed to just expectation values. My impression however is that this in itself is maybe not a major technical complication. As long as the quantity to be bounded depends on the oscillating frequencies one can likely apply the result of [42] without a lot of technical difficulty.

In that sense, the paper can perhaps be seen as a straightforward generalization of [42] (which, to the author’s credit, is appropriately acknowledged throughout the manuscript). It may be that considering these multi-time processes is a conceptually non-trivial point, but the authors do not really make it clear enough in which type of situations this could be particularly interesting.

In this setting, in which it is important to consider measurements at different times, why should we care about processes that have equilibrated? Is it because we want to know whether all those different measurements will yield the same outcome? Is it just because it allows us to get other figures of merit? That is, I do not find the motivation for merging of equilibration results and the process tensor formalism strong enough. Considering all this, I do not quite find that the paper satisfies the journal’s acceptance criteria.

I also do not understand why they decided to split [14] and this reference into two. The setting of both papers appears to be the same, the difference being that the present paper considers finite time bounds, as well as slightly more general figures of merit. I do not find the two sets of results different enough to grant two papers, and together they could have perhaps made for a slightly stronger message: that analytical equilibration bounds generalize to multi-time processes, as opposed to doing the “multi-time process analogue” of each analytical equilibration result separately. In any case this is likely better left to the author’s judgment, they might have reasons to do so.

Another important point is that [42] (or for that matter, any other general analytical attempt to that problem) is rather limited in practice. The authors here mention that the bound grows with the system dimension as log d, which can be easily compensated by the fact that the effective dimension scales like d (note that in a many body system d is exponential in system size). The problem, which the authors may have missed (or at least I have not seen discussed), is that the biggest issue with the bound in [42] is the quantity N(\epsilon), which counts the number of energy gaps in a given interval.

In a many body system with a typical spectrum, this number is expected to be exponential in system size, likely down to epsilons that are exponentially small in system size. To give a rough idea, in a many body system one has to “fit” exponentially many levels in a spectrum of polynomial width. Thus there are inevitably very narrow energy windows with exponentially many energy levels. The same discussion also holds for the energy gaps.

As such, it is very likely that the equilibration time predicted by [42] is exponential in system size. As the authors discuss, this is still likely much smaller than the recurrence time, which is something, but it is still a big overestimate of the equilibration timescales in most models. In “real systems” these are expected to be perhaps polynomial in system size (which can be understood, for instance, in terms of transport and the relaxation of hydrodynamic modes).

Other comments:

-Why can you not apply Chebyshev’s inequality to the result? I think you can, just not writing it in terms of distance to the average over [0,T] but to the long time average.

-If the equilibrated process is not “classical”, what is the significance of the distance to classicality bein equilibrated? This point towards the end, and its actual significance, does not seem very well explained, and from the discussion it is perhaps a bit difficult to understand the notion of classicality clearly.

If the authors think there is room for it, I think the paper could do with a better motivation for considering this setup, and a clearer explanation of the novelty with respect to previous works on equilibration (if there is). I cannot think of specific changes beyond that.