Abstract
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^4ϕ4 theory, chiral perturbation theory (\chi PTχPT) 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 PTχPT, 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 \rhoρ 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 PTχPT Adler zeros. We perform a preliminary analysis to constrain the allowed space further, using ideas involving positivity inside the extended Mandelstam region, and other quantum information theoretic measures based on entanglement in isospin.
Highlights
In fig.(1) this is indicated by P. By construction, this still leaves behind a huge set of potentially interesting S-matrices. This begs the question: Can we distinguish these S-matrices, all of which lead to similar scattering lengths? At leading order in sophistication for instance, which of these S-matrices is closest to 1-loop χ P T –can we use hypothesis testing to answer this? it raises the question: Can we reduce the set of S-matrices by not imposing the experimental scattering lengths?
The region we find is indicated in fig.(1)–we call this the “River” since the figure is very suggestive of one with two banks! We find this region by imposing the ρ-resonance2 and the inequalities suggested by χ P T and the D-wave dispersion relations—equivalently the definite sign conditions on relative entropy referred to above
There is a region on the other bank which is far (in the sense of the (s0, s2) values) from the perturbative 1-loop region, which gives rise to scattering lengths comparable to experiments. This second region admits a set of reduced density matrices which close to the threshold cannot be distinguished from the other green region close to the 1-loop χ P T point, and exhibits comparable entanglement
Summary
It is often a worthwhile pursuit to tackle old problems using new tools. In this paper, we will consider the very standard 2-2 scattering in quantum field theory using certain tools in quantum information theory. In 2-2 scattering we can take ρ1 and ρ2 to be the reduced density matrices corresponding to one of the outgoing particles reaching detectors placed at certain angles (cos θ = x) in the centre of mass frame In such a scenario, where we consider Gaussian detectors of width σ and small angular separation ∆x, we will be able to show that. We use relative entropy considerations to study S-matrix bootstrap constraining pion scattering. There is a region on the other bank which is far (in the sense of the (s0, s2) values) from the perturbative 1-loop region, which gives rise to scattering lengths comparable to experiments Put differently, this second region admits a set of reduced density matrices which close to the threshold cannot be distinguished from the other green region close to the 1-loop χ P T point, and exhibits comparable entanglement. The green ring corresponds to this angular spread of ∆α
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