Non-Markovian noise that cannot be dynamically decoupled

Reply to Report 1:
The referee appears to be mostly happy with the paper. We removed Fig. 2 as requested. Regarding the statement that our model is a toy model, see our reply to Report 2.


We provide a line-by-line rebuttal of the critique in Report 2.

“Looks like the numerical analysis has been included to make up for the lack of a solid mathematical result”. We would like to point out that numerical analysis is a recognised and solid mathematical discipline. It is common practice in theoretical physics to use numerics whenever exact proofs appear to be out of reach, and that is exactly what we did: we went analytically as far as possible and then continued numerically. We also included the source code so that the sceptical reader may verify the numerics for themselves. This makes the paper mathematically complete and solid.

“The counter-example is highly artificial.” Counter-examples to folklore statements tend to be less obvious, less natural and less intuitive - otherwise the folklore would not have arisen in the first place. We agree that the statement in consideration is probably true for a number of common models but our aim here is to show that it is not always true; hence one will have to be careful when using the statement and it will make sense to determine more refined conditions in future as to when it does hold. That said, our model is a mathematical toy model that fulfills all requirements of a quantum mechanical model and it is hence sufficient to contradict the folklore statement. Whether a model is relevant or physical is a rather subjective question and the task of “finding a relevant counter-example” is not at all well-defined. We are not in any way claiming that one will find our counterexample “in general” in the lab, although we showed in the magnetic picture that the Hamiltonian shares common features with those encountered, and we hope that our discussion in Section 5 about the possible addition of local bath and system terms will make you more convinced of its physicality. The singular features of our Hamiltonian are also shared amongst most of the models used in quantum open systems. So our counterexample can stand as a warning: here’s a new feature which people were previously unaware of why DD can fail. Here’s an example which shows that one has to be careful with lots of bread-and-butter techniques in open system and control (spectral density, perturbative treatments, bath-system decomposition) when there are unbounded baths. It also should be considered as a starting point and guidance - what features (additional hypothesis) would future proofs have to have so that the folk-knowledge is restored?


“The behavior displayed is not what practitioners using dynamical decoupling schemes would ever worry about. In fact, they would never consider the infinitely fast limit scheme which is the main mathematical focus here.” Practitioners in dynamical decoupling are interested in high decoupling pulse frequencies - typically, the higher the frequency the smaller the decoupling error, and the fundamental idea of dynamical decoupling is that the decoupling error should converge to 0 as the pulse frequency tends to infinity. If decoupling does not work in this limit then it won’t work for arbitrary finite frequencies either, so a statement about the limit is actually stronger and mathematically more elegant. Also, we would like to point out that in practice when working numerically, we always deal with finite frequencies, which is exactly what would happen in the lab. Hence the behaviour we study is highly relevant.

“The discussion over the counter-example and its significance is convoluted.” The referee does not provide any justification for this statement and it is unclear what could be changed about the exposition to address this concern.

“No evidence that this is actually a "folklore".” This is not true. On p.2 of the original version we introduced the statement and provided references [13,14]. [14] is a review article. In order to strengthen this point, we added further exemplary references [15,16,17] in the revised version. We have also replaced “folklore” with the more appropriate term “mechanism” in the manuscript.

“The non-Markovian model considered here is trivially a single particle”. Notice that despite its simplicity, the Lee Friedrichs model is fully Markovian even in the single excitation sector. Moreover, our model is also infinite dimensional (although not with infinitely many modes). The referee writes that we chose a non-Markovian model due to it being “flashy”. This is incorrect. It is already known that Markovian models cannot be decoupled, so to go beyond the state of the art we had to consider non-Markovian ones. We agree with the referee that Markovian dilations tend to be mathematically much more complex than non-Markovian ones. It is exactly in this spirit that it is surprising that even simple models such as ours cannot be decoupled. The purpose of the paper is to emphasise this to the community which believes otherwise (see above).

“The authors mention non-markovianity, but it is not clear what they mean by this.” This is not true, we did define it (comment [9]). This is clearly non-Markovian according to most measures, although some authors (including ourselves in other contexts) would also consider certain models (e.g, the Shallow Pocket model) with exponential decay non-Markovian.

“I am particularly puzzled by the following bizarre phrase in the conclusion...” What we meant by our original sentence is that our model should not be classified as pathological and hence be dismissed but that there may be additional meaningful hypotheses which could be added in order to exclude counterexamples such as ours, and that this is an important future line of studies. We have now changed that sentence in order to be less contentious.

Changes we made to the manuscript:
* We added more references in Section 1 to show that the Zeno effect is indeed used as a mechanism to explain dynamical decoupling and have replaced “folklore” with “mechanism”.

* We rephrased the sentence in Section 5 that Referee 2 had flagged up.

* We removed Fig. 2 as requested by Referee 1.