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Relative entropy in scattering and the S-matrix bootstrap

by Anjishnu Bose, Parthiv Haldar, Aninda Sinha, Pritish Sinha and Shaswat S Tiwari

Submission summary

As Contributors: Parthiv Haldar
Preprint link: scipost_202008_00006v1
Date submitted: 2020-08-11 14:59
Submitted by: Haldar, Parthiv
Submitted to: SciPost Physics
Discipline: Physics
Subject area: High-Energy Physics - Theory
Approach: Theoretical


We consider entanglement measures in 2-2 scattering in quantum field theories, focusing on relative entropy which distinguishes two different density matrices. Relative entropy is investigated in several cases which include $\phi^4$ theory, chiral perturbation theory ($\chi PT$) describing pion scattering and dilaton scattering in type II superstring theory. We derive a high energy bound on the relative entropy using known bounds on the elastic differential cross-sections in massive QFTs. In $\chi PT$, relative entropy close to threshold has simple expressions in terms of ratios of scattering lengths. Definite sign properties are found for the relative entropy which are over and above the usual positivity of relative entropy in certain cases. We then turn to the recent numerical investigations of the S-matrix bootstrap in the context of pion scattering. By imposing these sign constraints and the $\rho$ resonance, we find restrictions on the allowed S-matrices. By performing hypothesis testing using relative entropy, we isolate two sets of S-matrices living on the boundary which give scattering lengths comparable to experiments but one of which is far from the 1-loop $\chi PT$ Adler zeros. We perform a preliminary analysis to constrain the allowed space further, using ideas involving positivity inside the extended Mandelstam region, and elastic unitarity.

Current status:
Editor-in-charge assigned

Submission & Refereeing History

Reports on this Submission

Anonymous Report 2 on 2020-9-18 Invited Report


1- The idea is original and has many potential applications.
2- Many examples in the form of known theories are given.
3- Most impressively, the presented methods provide new constraints for the S-matrix bootstrap.


1- The appearance of the square of the delta function and how it is handled is not really convincing. However, it seems that this cannot be avoided and the obtained results are convincing.
2- It seems that while conditions (5.10) on scattering lengths are rigorously derived, the conditions (5.11) are merely assumed based on two examples (experiments and chiral perturbation theory). Following the bootstrap philosophy of constraining the space of theories on theoretical grounds, it would make sense to also discuss the improvements to the S-matrix bootstrap that follow from (5.10) alone, without imposing (5.11).
3- The name "river" for figure 1 is a bit unfortunate and might be confusing as it turns around the established association of land with allowed regions (lake, peninsula, island, archipelago, ...). "Land bridge" might have been more fitting.


The paper is well written, provides many details and exposes the idea of studying scattering using relative entropy from many angles. It meets the expectations of the journal as it opens a new pathway into the S-matrix bootstrap and provides a novel link between scattering theory and quantum information theory. The general acceptance criteria of the journal are all met. I recommend publication in SciPost Physics after a very minor revision.

Requested changes

I would leave it up to the authors if and to what extend the points under weaknesses are addressed. The following small typos should be corrected:

1- Two of the \rho_1 above (3.8) should be \rho_2.
2- Fix arguments of delta functions in (C.2).
3- Below (C.30) there should be an equality for I_g.

  • validity: high
  • significance: top
  • originality: top
  • clarity: high
  • formatting: excellent
  • grammar: good

Anonymous Report 1 on 2020-9-11 Invited Report


1. Original idea and interesting results
2. Detailed calculations


1. Structure of the paper is difficult to follow
2. Missing more physical discussions of the results


This paper studies how entanglement can be used to constrain the space of valid S-matrices. The authors study the relative entropy between two density matrices which can be interpreted as two different theories or the same theory being measured by a Gaussian detector. The authors go one step further by connecting the relative entanglement entropy with the usual positivity bounds for effective field theories. Moreover, assuming the positivity bounds, they show that the positivity of the relative entanglement can further constrain the space of S-matrices. In particular, it can exclude a big part of the parameter space without the need of experimental inputs.

This is a new and interesting direction and indeed deserves further exploration. Before recommending publication, I would like to ask a few clarifications from the authors.

Requested changes

1. Below Eq. (2.2) the authors write that the analysis differs in crucial points from the ones in Refs. [3]. However, these crucial points are not explained with much detail and I believe it is worth a better comparison with the previous literature. Their main point is that instead of integrating over the full phase space and having to deal with divergences, they integrate just over the part in which the detector is sensible. Then, calculating the relative entropy, they are able to get rid of the detector dependence at least at leading order. However, what is not obvious is how problematic is the regularization issue, that the authors claim to avoid, as the regularization scheme just modifies an overall factor (PRD 100 (2019) 7, 076012 ) and in the end it is not important when looking to the entanglement variations.

2. One point raised in the previous literature is that the density matrix is not Lorentz invariant if one looks to only the spin or momentum entanglement (see e.g PRD 97, 016011 (2018)). As the authors are always looking just to the momentum entanglement, why this is not an issue for the spinning generalization in sec. 2.3 ?

3. The S-matrix is defined as S=1+iT and, of course, the interesting part is in the T-matrix. However, the 1 part should be taken into account in the normalization of the density matrix. It is not clear if this is the case.

4. Regarding the processes in Eq. (4.11), the ones in the first and last lines have the same initial/final Hilbert space, which seems to not be the case for the process in the second line. I wonder if makes sense to compute the entanglement entropy in this case. The authors could clarify this point and also add an explanation in the text.

5. The fact that positivity bounds from the relative entanglement is correlated with the usual positivity bounds, as in Eq. (4.25), seems related with the fact that the relevant quantity in both cases is the second derivative of the amplitude. However, a more physical explanation is missing in the text.

6. Eq. (5.25) implies that the maximum of the relative entanglement is in the forward region. Is there a physical reason? Moreover, this seems to not be the case for the black/red curves of Fig. 5. Why?

7. In general, the relative entropy is not only non-negative but also bounded from below (Rev.Mod.Phys. 90 (2018) 3, 035007). Could this have any impact in the results, giving stronger constrains than just imposing positivity?

Moreover, a few improvements in the paper structure would be desirable. For example, in pag. 7 it is confusing that the angles $\theta_{Dt}$, $\theta_D$ and $\alpha$ are the same, but it would be more convenient to unify the notation for $\theta_{Dt}$ when referring to the detector angle (a similar confusion also appears in other parts). Also, $x$ is occasionally used as $\theta$ (e.g. fig. 3) but it is actually the $\cos\theta$. Section 3.2 seems to consider the case A+B$\rightarrow$A+B but fig.3 refers to A+B$\rightarrow$C+D, which is then only discussed in sec. 3.5. Another confusing point is the $F$ and $g$ functions, as they are basically the same and the only important point is that the support function is chosen as a density probability function in order to ensure the positivity properties of $\mathcal{P}_g$. One could emphasize this point in the definition of Eq. (2.4). In Eq. (3.1) the replica trick is shown and not the definition of the entanglement entropy. For someone who is not familiar with the geographic plots - river, lake, peninsula - it would be helpful to add in the captions that the blue is the allowed region.

  • validity: high
  • significance: high
  • originality: high
  • clarity: good
  • formatting: good
  • grammar: good

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