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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

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Submission summary

Authors (as registered SciPost users): Parthiv Haldar · Aninda Sinha
Submission information
Preprint Link: scipost_202008_00006v2  (pdf)
Date accepted: 2020-11-13
Date submitted: 2020-10-13 09:52
Submitted by: Haldar, Parthiv
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
  • High-Energy Physics - Theory
  • Quantum Physics
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.

Published as SciPost Phys. 9, 081 (2020)

List of changes

List of Changes to the Draft
To address various points raised in invited referee reports, we did some significant changes to the draft. We list out these in the following.
1. We added an analysis of entanglement in isospin for pions in section 8 supplemented by Appendix E. We explore quantum relative entropy and a quantity called entanglement power. In Appendix E.2, we give a concise review of the basic concept of entanglement power for the convenience of the reader as well as to make the presentation self-contained. In the main text, i.e. in section 8, we provide results of some elementary numerical analysis. We report some preliminary findings from the numerical exploration in regards to entanglement power in the “Future Directions” section, which we wish to present in future work.

2. We shifted the formal setup for analysis involving spinning particle entirely to the appendix, removing the corresponding section from the main text. Since, in this work we consider only entanglement in momentum space and isospin space for spinless particle, we thought it to be judicious to keep the explorations for the spinning particle to the appendix. The appendix serves as the formal point of invitation for future exploration. As rightly pointed out by the honourable, the issue of Lorentz invariance of entanglement involving spinning one is a significant one. We wish to address this fully in future endeavour.

3. Corrected an issue in the definition of $D_Q$ in eq.(3.19). The $D_Q$ does not depend upon the human chosen parameter $\sigma$ and is universal.

4. Added a footnote on page 10 (footnote 3) to address the confusion surrounding the notation of $F$ and $g$ around equation (2.14) . We would like to point out that $F$ is the function entering the projector $Q_{AB}^{(F)}$ defined in eq.(2.4). $F$ is assumed to be a function of $\theta$ without any special form assumed. In the final density matrix, it is $F^2$, and not $F$, that enters. We have used $g$ to denote $F^2$. Further, at this point we have assumed that $F$ is a function of $\theta_A$ via $\cos\theta_A$, i.e. $F(\theta_A)\equiv F(\cos\theta_A)$. Denoting $\cos\theta_A$ by $x$, we have defined $g(x):=F(-x)^2$. This notation follows to all subsequent references to $F$ and $g$. Thus, mathematically, $F$ and $g$ are not same.

4. We made changes to both figure 2 and figure 3 to clear out confusions.

5. We fixed various typos. Further, we thoroughly checked the spelling and grammatical constructs using Grammarly.

Reports on this Submission

Anonymous Report 3 on 2020-11-2 (Invited Report)


I recommend the publication of this paper.

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Anonymous Report 2 on 2020-10-30 (Invited Report)


All the changes requested by me were implemented, as well as several larger changes requested by the other referees. I recommend the paper in this form for publication.

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Report 1 by Joao Penedones on 2020-10-26 (Invited Report)

  • Cite as: Joao Penedones, Report on arXiv:scipost_202008_00006v2, delivered 2020-10-26, doi: 10.21468/SciPost.Report.2114




The new version of the paper successfully addresses most of the comments raised by the referees. In particular, the new discussion about entanglement in isospin space is a very significant improvement of the paper. Therefore, I recommend the paper for publication.

I have only one minor question: is it possible to simplify the expression for entanglement power and write it solely in terms of the phase shifts? A similar result was obtained in equation (3) of reference [42] for the case of nucleon scattering.

Equation (E.11) has two typos (the second $\beta_1$ in the state and $b_2$ in the exponent should both be $\beta_2$).

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Author:  Parthiv Haldar  on 2020-11-06  [id 1036]

(in reply to Report 1 by Joao Penedones on 2020-10-26)

We had looked into an analytic formula for the entanglement power, $\mathcal{E}$, akin to equation (3) of reference[42]. But, the problem is that there is a complicated angle dependence in the denominator that makes analytic evaluation of the final two angular integrals not possible (at least we have not managed it).

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Anonymous on 2020-11-06  [id 1037]

We have looked into an analytic expression for the entanglement power $\mathcal{E}$ akin to equation (3) of reference[42]. But, the problem is that there is a complicated angle dependence in the denominator that makes analytic evaluation of the final two angular integrals not possible (at least we have not managed it).