A perturbation theory for multi-time correlation functions in open quantum systems

I would like to thank the Reviewers for their careful reading of the manuscript and for providing positive recommendations. Detailed responses to the Reviewers' questions and comments are given below.

Yours sincerely, P. Szańkowski

Comment 1

The manuscript is essentially missing an actual introduction about multi-time correlation functions in open quantum systems, the quantum regression formula, and also the attempts to go beyond it — as said, there are not many, but indeed there are some [...].

I thus recommend to include an actual introduction with the content specified above and some references (those above, and/or others that can be useful) and only after that to start with the argument about the propagation of thermalization, clarifying that it is a — interesting and physically relevant — case study to clarify the limits of applicability of the QRF; this part might even be written in a separated section.

Answer: I sympathize with the sentiment expressed in this comment; the structure of the paper does indeed deviate from the standard pattern. In fact, the initial drafts of this paper were written along such, more traditional, lines: The Introduction had a form suggested in the comment, and in content it more-or-less mirrored the current abstract. Then, I followed with sections containing technical definitions of MTC, first in closed, and then in open systems. Actually, back then, the problem of thermalization propagation—its introduction and solution via just defined perturbation theory—was crammed into a single section at the end of the paper.

However, it became quite apparent to me—and my feelings were validated by the feedback I got—that the traditional pattern does not work for this paper in particular, and maybe even the topic in general. The issue is that the target audience of this kind of work (i.e., MTCs plus open systems) is composed of two, non-overlapping, camps. On the one hand, the language of MTCs is natural for people with background in field theories (e.g., condnsed matter physics, ultra-cold gases, quantum optics, etc.), for many of whom the open systems framework is rather foreign. On the other hand, we have the portion of the audience "lured" in by the open systems aspect, who typically come from quantum information-adjacent backgrounds, where it is the concept of MTCs that is rather unnatural. (It is quite remarkable how often I had to explain to my colleagues that MTCs and correlations between sequential measurement results are not the same thing.) Therefore, to construct a proper introduction, which would not alienate either part of the target audience, one cannot rely on the reader's familiarity with MTCs nor the concepts of the standard open system theory—in other words, not only you have to define all the elements (MTCs, dynamical maps, GKLS theorem etc.) but you also have to show examples how those things are useful. Integrating this kind of introduction with the narrative about thermalization and its propagation—which is a topic universally familiar to physicists of any background—is an efficient way to bestow the otherwise technical introduction with a fairly easy to follow structure that showcases the use case for every concept involved. Hopefully, I was able to make my case why I prefer to keep the structure of paper as it is now.

In regards to the references pointed out by the Reviewer: upon reflection, I agree it was a mistake on my part that I neglected to cite some of the previous works concerning extensions to quantum regression theorem, which is, of course, strongly linked with open system MTCs. I was familiar with a couple of those works (although, not all of them, thank you for bringing them to my attention), but I decided that the approach adopted by them diverges too much from what I was doing in my paper. However, now I see that this difference in approach is a good point in and of itself—I should have reevaluated my assessment. In response, I have added the following paragraphs to the conclusions:

Previous works—e.g., [deVega_PRL2005,deVega_PRA2006,deVega_PRA2007,McCutcheon_PRA2016]—have proposed approximation schemes to improve on the QRF method. The approach adopted in these works is, in a sense, complementary to ours. The standard strategy is to address the problem of MTCs by extending the so-called quantum regression theorem (QRT) beyond the zero-correlation-time regime. Essentially, the QRT states that when $\tau^e \to 0$, the single- and two-time correlations of system observables satisfy the same dynamical equation [McCutcheon_PRA2016]; hence, QRT implies the exactness of QRF for zero-range cumulants, which is, of course, consistent with our theory. "Extending QRT" then amounts to writing down the derivative of a two-time MTC with respect to the second time argument and proposing an approximation that turns the expression into a legitimate dynamical equation—that is, closing the equation by reassembling the MTC on the right-hand side. Such an approximation is necessary because, as per QRT, the equation does not close on its own when $\tau^e > 0$.

It follows that theories utilizing this dynamical-equation approach result in MTC approximations analogous in form to the QRF method, where there are no explicit connections reaching across interventions. Instead, the deployed approximations lead to effective propagators between interventions that attempt to capture the effects that would result from cross-intervention connections, were they actually included. In contrast, our approach facilitates such cross-intervention connections, thereby allowing for the independent treatment of the in-between propagators, as demonstrated in Sec.5. Seemingly, there is no decisive argument favoring our approach over the traditional equation-of-motion method. When faced with a concrete problem, one must carefully consider the pros and cons of each method, and which approach is preferable will most likely depend on the fine details. However, there is something to be said for how intuitive it is to build up corrections to the MTC by progressively including more complex temporal correlations connecting the propagators across interventions—no such clear picture can be used to illustrate the physics of perturbation theory within the equation-of-motion framework.

Comment 2

At page 7, after equation 2.13, the term "discrete-time combs" appears, but without clarifying what the name "combs" is referring to. In view of the widespread use of the term "comb" within the community of open quantum systems — see e.g. Phys. Rev. Lett. 101, 060401 (2008) or the recent review arXiv:2503.09693, it should be clarified whether the name "comb" is referring to the same objects and the differences/peculiarities should be specified.

Answer: I have changed the name to "discrete-time filter function" to avoid confusion with quantum combs. Originally, I had "Dirac comb" in mind when I was naming the function $\Delta$; to be frank, I completely missed the conflict with process tensors/quantum combs. Thank you for pointing this out to me.

Comment 3

Does the notation in the second equation in (5.12) mean that $\Lambda_{t',t'}$ is the identity map? Maybe a more explicit notation or at least a clarification of this would help the reader.

Answer: Indeed, the standalone "bullet" $\bullet$ correspond to super-operator identity. I have added the line

where the super-operator identity is expressed using the "bullet" notation, $\bullet\hat A = \hat A$.

below Eq. (5.12) to clarify this notation.

Comment 4

According to the general criteria of SciPost Physics, the conclusions of the manuscript are missing the "perspectives for future work".

Answer: To address this issue I have added the following paragraph at the end of Sec. 8 (Conclusions):

Finally, we note that the hierarchy of correlation strength (3.6), which constitutes the basis for our theory, can be analyzed in terms of graphs. In this reading, one interpretation is to represent the moments of dynamical variables as the graph's vertices, and the edges connecting vertices as cumulants. Therefore, the introduced perturbation theory should naturally lend itself to some form of diagrammatic treatment. This opens a promising avenue for further development of the theory, with the prospect of a high payoff that is characteristic of diagrammatic methods known from other perturbation theories compatible with such an approach.

I would like to thank the Reviewer for their careful reading of the manuscript and for providing positive recommendations. Detailed responses to the Reviewers' questions and comments are given below.

Yours sincerely, P. Szańkowski

Comment 1

While the Introduction explains the problem with great clarity, the abstract does not. It says little about what is gained by computing multi-time correlations.

Answer: I have modified the abstract with a short list of topics where multi-time correlations play a crucial role.

Comment 2

It might help the reader to outline the structure of the paper at the end of the introduction.

Answer: I have added the outline at the end of the introduction.

Comment 3

In Sec. 2, the author begins with the structure of multi-time correlations in quantum physics by introducing the notion of bitrajectories. The author is completely right about dynamical maps' inability to describe multi-time correlations; however, they can be generalized to deal with multi-time correlations using quantum combs or the process tensor, see PRXQuantum.2.030201. See arXiv:0712.0320, arXiv:1712.02589, arXiv:1802.03190, and arXiv:2304.10258 on notions related to bitrajectories. Also, see a recent review on these methods at arXiv:2503.09693.

[...] I do think that the author should make links to the process tensor formalism, as this community is looking at multi-time quantum correlations from many different perspectives, and this will also give this paper the exposure that it deserves.

Answer: I thank the Reviewer for this comment. While developing the bi-trajectory language, we realized some time ago that there might be some links between the formalism we were working on and the process tensor approach. Soon after, we concluded that this link is real and it most likely runs much deeper than we initially thought. In Quantum 8, 1447 (2024) we wrote about our preliminary findings about these relations; it is clear to me that there is much more to this story. At the time of writing this paper it did not feel right to me to point out the fundamental connections between combs and bi-trajectories, without first researching the topic in more detail. Moreover, it seemed at this time that the comb connection might be the next full project we will be doing with my co-workers—it would be a waste to spoil our future findings with half-baked preliminary conclusions we had so far, I thought. (Plans got delayed, but we are still very much interested in pursuing this topic.) Of course, now I see that omitting quantum combs for such reasons was a silly mistake. To remedy this I have added a comment (and relevant references) in the opening paragraph of Sec. 2.2:

Multi-time correlation functions are ostensibly dynamical quantities. As such, the formal description of MTCs in terms of the standard state-focused formulation of quantum mechanics---which is tailored for describing `quantum statics' (i.e., single-time MTCs, or simply, expectation values of observables)---is a sub-optimal choice. Recently renewed interest in contexts extending beyond the standard quantum statics has stimulated development of powerful tools such as process tensors (or quantum combs) [Milz_Quantum2020, Milz_PRXQuantum2021, Taranto_arxiv2025], that can efficiently handle various multi-time objects encountered in quantum mechanics. Here, our method of choice is the so-called bi-trajectory formalism; it was developed previously in [Szankowski_SciRep2020, Szankowski_PRA2021, Szankowski_Quantum2024, Lonigro_Quantum2024, Szankowski_arXiv2024] and it is a natural fit for the specific context of open system MTCs. The bi-trajectory theory is a reformulation of quantum mechanics that shares some of its DNA with other trajectory-focused formalisms [Caves_PRD1986, Aharonov_PRA2009, Strasberg_PRX2024]. Moreover, a limited preliminary investigation [Lonigro_Quantum2024] suggests a deeply rooted connection between bi-trajectory formalism and process tensors, albeit the full understanding of this potentiality requires further study.

Comment 4

My guess is that Eq. (3.5) cannot be exact due to the structure of quantum Markov order, see PhysRevLett.122.140401. However, an approximate versions should hold due to npj Quant. Info 7, 149 (2021).

Answer: This comment seems to suggest it cannot be true that correlation functions/cumulants have finite range (and thus, finite correlation time) because typical quantum systems have infinite Markov order. If I am not misreading its meaning, the comment conflates the correlation time with the Markov order (understood, loosely, as the number of past conditions influencing the future). The implication is that short correlation time should correspond to a small number of past conditions, and long correlation time to a large number of conditions; in particular, Markovian processes (Markov order equal 1) would have to have shorter correlation times (zero?) than non-Markovian processes (Markov order greater than 1).

This, however, is incorrect: in reality, the range of correlations/cumulants is independent of the notion of Markovianity, Markov order, or other related concepts. It is a very common misconception that they are connected (or even the same), and, sadly, the literature is plagued by it. I think the unfortunate choice to dub Markovian processes as "memoryless" is largely to blame. Also, it does not help that the most common way to construct Markovian noise is to drive an otherwise deterministic system with a zero-correlation time white noise—this, I think, can inadvertently create a strong association in people’s minds that short correlation time leads to Markovianity, even though the two properties belong to two different processes (the driver is white but can be non-Markovian, cf. Phys. Rev. E 50, 2668 (1994) and the driven is Markovian but is non-white).

I find the following rule of thumb to be helpful in understanding the difference between Markovianity-related concepts and the correlation time: the "memorylessness" (i.e., Markovianity etc.) tells us how the process is "made", while correlations/cumulants tell us how the process "looks like from the outside". For example, in the context of open systems, the dynamics of the system–environment coupling operator is what directly influences the system. Therefore, the influence from the environment is encapsulated by correlation functions of the coupling. Since, in general, the same correlation functions can be produced by different underlying dynamical laws governing the environment, it is irrelevant whether the environment can or cannot be considered as Markovian.

Also notice that for Markovian processes (or a process with larger Markov order), the time-scales associated with the past conditions are completely irrelevant: the timings of the events constituting the process history do not matter, the only thing the Markovian process cares about is the latest event. Even if past events are clustered infinitesimally close to each other, or if the last event in the sequence happened in the distant past, Markovian process always depends only on the latest event. This focus on the relative chronology over the actual timing of past conditions only underlines the disconnect between Markovianity (Markov order in general) and time-scales associated with the process dynamics. Of course, in concrete calculations, the timings of the past conditions does affect the value of correlation functions and other derived quantities, regardless if the process is Markovian or not.

A classic example illustrating these points is a deterministic process describing a planet orbiting its sun: the process representing planet's position is Markovian (it is governed by time-local equations of motion), but the correlation time is infinite (or undefined, depending on one’s preference) because correlation functions of position are practically periodic.

It is not difficult to construct an analogous quantum model: Typical arrangement constitutes a system with discrete spectrum, $\hat H = \sum_a h(a)|a\rangle\langle a|$ and stationary initial state, $\hat\rho_0 = \sum_a p(a)|a\rangle\langle a|$. The system is Markovian by any metric: (i) its evolution $\hat U_{t,s} = {\exp}[-\mathrm i(t-s)\hat H]$ is CP divisible, $\hat U_{t,s}\hat U_{s,0} = \hat U_{t,0}$; (ii) quantum regression is exact, etc. Nevertheless, the correlation time is infinite (or undefined), as correlations of system’s observables are oscillatory, e.g.,

However, when the system is complex, so that its spectrum becomes dense and degenerate, the revivals of the oscillations in correlations of some observables are getting more and more delayed. At some point, if this delay is long enough, one can treat correlation as a decaying function with finite correlation time (Phys. Rev. A 100, 052115 (2019))—this is the standard physical argument why environments like thermal baths should have finite-range correlation functions. Notice that the system remains Markovian: the Hamiltonian gets more complex, but it still generates divisible evolution.

In summary, Eq.(3.5) is inexact in the sense that it assumes that the recurrence time is so much longer than any other time-scale so that the system's correlations/cumulants can be treated as decaying rather than oscillatory functions. This assumption applies to sufficiently complex systems and its validity is unaffected by the Markov order characterizing the system.