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.

中文翻译：

---

散射中的相对熵和S矩阵自举

我们在量子场论中考虑2-2散射中的纠缠度量，重点是区分两个不同密度矩阵的相对熵。在几种情况下研究了相对熵，包括$ phi ^ 4 $理论，描述II型超弦理论中的π散射和dilaton散射的手性摄动理论（$ \ chi PT $）。我们使用大量QFT中弹性微分截面上的已知范围，得出相对熵的高能范围。在$ \ chi PT $中，接近阈值的相对熵在散射长度的比率方面具有简单的表达式。在某些情况下，相对熵的确定符号性质超过了相对熵的通常正性。然后，我们转向在pion散射情况下对S矩阵自举的最新数值研究。通过施加这些符号约束和$ \ rho $共振，我们发现了对允许的S矩阵的限制。通过使用相对熵进行假设检验，我们分离了生活在边界上的两组S矩阵，它们给出的散射长度与实验相当，但其中一组距离1环$ \ chi PT $ Adler零点很远。我们使用涉及扩展Mandelstam区域内的正性和弹性统一性的思想进行初步分析，以进一步限制允许的空间。我们分离出了生活在边界上的两组S矩阵，它们给出的散射长度与实验相当，但其中一组距离1环$ \ chi PT $ Adler零点很远。我们使用涉及扩展Mandelstam区域内的正性和弹性统一性的思想进行初步分析，以进一步限制允许的空间。我们分离出了生活在边界上的两组S矩阵，它们给出的散射长度与实验相当，但其中一组距离1环$ \ chi PT $ Adler零点很远。我们使用涉及扩展Mandelstam区域内的正性和弹性统一性的思想进行初步分析，以进一步限制允许的空间。