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Relative entropy in scattering and the Smatrix bootstrap
by Anjishnu Bose, Parthiv Haldar, Aninda Sinha, Pritish Sinha and Shaswat S Tiwari
This Submission thread is now published as SciPost Phys. 9, 081 (2020)
Submission summary
As Contributors:  Parthiv Haldar · Aninda Sinha 
Preprint link:  scipost_202008_00006v2 
Date accepted:  20201113 
Date submitted:  20201013 09:52 
Submitted by:  Haldar, Parthiv 
Submitted to:  SciPost Physics 
Academic field:  Physics 
Specialties: 

Approach:  Theoretical 
Abstract
We consider entanglement measures in 22 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 crosssections 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 Smatrix bootstrap in the context of pion scattering. By imposing these sign constraints and the $\rho$ resonance, we find restrictions on the allowed Smatrices. By performing hypothesis testing using relative entropy, we isolate two sets of Smatrices living on the boundary which give scattering lengths comparable to experiments but one of which is far from the 1loop $\chi PT$ 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 selfcontained. 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.
Submission & Refereeing History
Published as SciPost Phys. 9, 081 (2020)
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Reports on this Submission
Anonymous Report 3 on 2020112 (Invited Report)
Report
I recommend the publication of this paper.
Anonymous Report 2 on 20201030 (Invited Report)
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.
Report 1 by Joao Penedones on 20201026 (Invited Report)
 Cite as: Joao Penedones, Report on arXiv:scipost_202008_00006v2, delivered 20201026, doi: 10.21468/SciPost.Report.2114
Strengths
1
Report
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 $\beta_2$).
Author: Parthiv Haldar on 20201106 [id 1036]
(in reply to Report 1 by Joao Penedones on 20201026)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).
Anonymous on 20201106 [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).