Relative entropic uncertainty relation for scalar quantum fields

1. The problem is very well motivated in the introduction. Quickly grab the reader's attention.
2. The computations are very clear and can be easily reproduced/verified.
3. The result is based on precise mathematical calculations, and does not involve "leap of faith" arguments.
4. It is very well written.

In addition to the major drawback that I point out in the section report, I would like to point out the following:

1. The result is "hidden" in the text. It forces the reader to go through a lot of technical details to get to it. It would be much better if the authors anticipated or referenced the result at the end of the introduction.
2. SUGGESTION: The authors justify the use of relative entropy to construct the uncertainty principle based primarily on the fact that relative entropy is finite at the limit of the continuum. But the same happens for difference of two von Neumann entropies. I think the authors could give better reasons for that choice. One of them is that the relative entropy can be defined and calculated (in principle) directly in the continuum QFT, without the need of introducing a cutoff (Araki formula).

In the present work the authors address the original problem of finding an entropic uncertainty principle for a scalar quantum field. Entropic certainty relations are a very important subject of study in modern physics because they have a very wide spectrum of applications: from quantum foundations to experiments. Contrary to what is often the case with finite quantum systems, where the von Neumann (vN) entropy is often used to formulate uncertainty relations, the authors uses the relative entropy. As it was pointed out for the authors, the main reason is that the relative entropy (for a correct choice of states) has a finite continuum limit, whereas the vN entropy does not. This allows them to report a result that holds even in the continuum QFT.

The main result of the work is the relative entropy uncertainty relation shown in equation (50). From my understanding on the subject, this expression is not on the same level as other known uncertainty relations because the r.h.s. still depends on the underlying states. What I expect for a uncertainty relation (like the one in eqs. (1) or (2)) is an inequality that indicates that the the sum of two entropic measures (or the product of two standard deviations) is greater or less than a bound which may depend on the theory but not on the state(s). I think the authors have to better argue in which sense their result can still be considered as an uncertainty relation at the same level as the other uncertainty relations that one can find in the literature. Or in which way one has to understand inequality (50) to affirm that the bound in the r.h.s. is still relevant and useful. Typically, the usefulness of an uncertainty relation is that even when one is not able to calculate the l.h.s, one can easily have a bound by just reading the r.h.s. This is not the case of the relation presented here.
Furthermore, according to eq. (66), it seems that eq. (50) can be understood as an improvement of the Robertson-Schrödinger uncertainty relation for the quantity that appears in the l.h.s. of (66). This would be another way of looking at the same result, but it still implies an uncertainty relation whose “bound” is state dependent.
I would consider this like a major revision. If the authors could address this problem, I would recommend this work for publication.

Some of these are requested changes, others are minor questions for the authors:

1. The sum in equation (3) would run from 1 to N to be compatible with the boundary condition \phi_0 = \phi_n.
2. In what sense eq. (5) is a unitary transformation? If I understood correctly, up this point the computation is in classical field theory. The quantization comes later.
3. In eq. (6), should it say \tilde{\phi} and \tilde{\pi}?
4. In eq. (8), the fields are written in capital letters contrary to what happened before. Presumably this change was made for the purpose of differentiating classical from quantum variables. The authors should stress that in the text.
5. For eqs. (14) and (15), it should be emphasized that \phi_l and \pi_l are real numbers, and that the basis are orthonormal.
6. I would like to ask the authors if they can provide a reference for equations (36-37). Otherwise, if it is a relation found by them, I would like to ask them about the relevant points in the calculation of such a relation.
7. From (50), can it be inferred that if a state has the same two-point functions as a coherent state, then it must be coherent?
8. It is not entirely clear to me what the authors are saying in the first paragraph of the 2nd column on page 9 (the one that starts with: “We begin by reformulating ...”). If one chooses a state that has the same two-point correlators as the reference state, then the the r.h.s. of (50) is identically zero. Is that correct?

1 - The computations are not particularly difficult and easy to reproduce.
2 - Conciseness.

1 - Not really an original work. Most of the results appear just as a multidimensional generalization of a single quantum harmonic oscillator.
2- The conclusions are not really emphasized, and one has to go through all the manuscript to understand what is the point of the whole discussion.

In this paper, the authors study the relative entropic uncertainty relation for scalar quantum fields. They find that relative measures of uncertaintes, comparing the state under analysis and a reference state, are well-behaved even in the continuum limit, meaning that they could be good notions for states of quantum field theories.
The work points out interesting observations in the context of quantum information and its relation with quantum field theory. While I do have some comments and few suggestions to improve the presentation, these are minor and I recommend publication.
Here is the list of comments and suggestion:

1- Page 5, after Eq. 39: "...with the thermal covariance being proportional to the vacuum one". This sentence does not really make sense; the two matrices are not proportional, since the "thermal" factor (1+2n_{BE}(\omega_l)) depends on the index. Maybe "...with the thermal covariance given by..." would be better.
2- Page 5, just before Eq. 42. "For a free theory in an equilibrium state, the mixed correlations... ... vanish". If I am right for "equilibrium state" the author means "invariant under time reversal", which would imply the relation < \phi \pi +\pi \phi> =0. The sentence is probably too concise to be understood at first glance and at least a reference for that is needed.
3 - Page 5, just after Eq. 42. "the eigenvalues of the correlator
product MN are at least the eigenvalues...". I do not really understand this sentence. In the ground state \bar{M}\bar{N} =1/4, so the only eigenvalue is 1/4; what does it means that the eigenvalues of MN are also eigenvalues of \bar{M}\bar{N} ?
4 - The conclusions of the work are hidden. The Eq. 50 is considered as the main result of the work, and it would be better to anticipate it in the introduction or emphasize it again in the conclusions.
5- It is not particularly clear, at least at first reading, what is the regime of validity of the results. While the Heisenberg relation or the "Bialynicki-Birula and Myciel ski" formula are generic, the focus of the manuscript appears to be restricted to certain class of states obtained as eigenstates or thermal states of the Klein Gordon theory. An improvement of the presentation would be the insertion of a precise statement (in the introduction) regarding the range of validity of certain inequalities.
6 - I would also suggest emphasizing more the connection with a previous work of the authors regarding the same topic (Relative entropic uncertainty relation, ref. [31]). Probably even a small comment/appendix summarizing the results available for a single random variable would be helpful.