Entanglement in Elastic and Inelastic Two-particle Scatterings at High Energy

We study the entanglement produced in transverse momentum by two-particle scattering at high energy. Employing the S-matrix framework for the derivation of reduced density matrices, we formulate the entanglement entropy for an inelastic scattering as well as an elastic one. We display the formulas of the entanglement entropy in terms of two-body cross sections. We also derive the entanglement density as a function of the transverse momentum. As an application, we then focus on both forward elastic ($pn \to pn$) and inelastic ($pn \to np$) channels scattering allowing for a fruitful comparison of the two reactions with the same proton-neutron content. We evaluate the elastic and inelastic entanglement entropy by using known parameterizations of experimental data for neutron-proton reactions. Comparing those entanglement entropies, we observe that the inelastic scattering produces more overall entanglement than the elastic one in the $pn$ sector.

💡 Research Summary

The paper investigates how much quantum entanglement is generated in the transverse‑momentum degrees of freedom during high‑energy two‑particle scattering. Using the S‑matrix formalism, the authors construct reduced density matrices for the final‑state particles and derive explicit expressions for the entanglement entropy (EE) of both elastic (pn → pn) and inelastic (pn → np) channels.

Key theoretical steps:

1. The total Hilbert space is split into the elastic two‑particle subspace H₁, a single inelastic two‑particle subspace H₂, and a multi‑particle sector H_X. The S‑matrix is written as S = 1 + 2iT, and unitarity yields relations among T‑matrix elements.
2. By projecting the S‑matrix onto the elastic and inelastic final states, the authors obtain the full density matrices ρ₁ and ρ₂. Tracing over one particle gives reduced density matrices ρ_{A₁} and ρ_{A₂}.
3. Because the momentum Hilbert space has infinite volume V = ∑_{ℓ}(2ℓ + 1), a divergent factor δ(0) appears. Following earlier work, they regularize this by “volume regularization,” effectively replacing δ(0)/V with a finite constant.
4. The Rényi entropy Tr(ρ_{A_i}ⁿ) is computed using a partial‑wave expansion of the scattering amplitudes τ_{ℓ}^{ij} and overlap functions f_{ℓ}^{ij}. In the limit n → 1 the von‑Neumann EE becomes

 S_{ij} = ln V² − ∫{−1}^{1} dcosθ P{ij}(cosθ) ln P_{ij}(cosθ),

where P_{ij}(cosθ) is a probability distribution built from the elastic and inelastic differential cross sections. For the elastic channel

 P_{11}(cosθ) = δ(1−cosθ)