Relative entropy in scattering and the S-matrix bootstrap

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.