  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.
