Scattering entanglement entropy between one particle species and the others

We construct the entanglement entropy of a general scattering process between one particle species (denoted as $A$) and the rest of the final state. Assuming a pure initial momentum eigenstate, we show that the reduced density matrix in particle $A$’s Fock space is block-diagonal in zero-, one- and multi-particle states, and is diagonal in the momentum basis for the one-particle state. As a result, we could write down the Von Neumann entropy of particle $A$, $S_A$, in terms of scattering probabilities and differential cross sections, assuming no more than one particle $A$ is produced in the final state. $S_A$ contains the information of both the total scattering probabilities (or cross sections) and the differential distributions — the latter depends on a phase-space resolution factor as a result of the infinite-dimensional momentum Hilbert space. Our result is general and model-independent since we consider the full final state from time evolution ($S\mathrm{\mid }i⟩$) rather than a specific outcome, and is applicable as long as the probability for producing multiple particle $A$ is subleading. We also obtain the $n\ge 2$ Tsallis and Rényi entropies in our framework and find out that, at the leading order, they are proportional to the total scattering probability and carry no differential information.