Entanglement and scattering in quantum electrodynamics: S-matrix information from an entangled spectator particle

We consider a general quantum field relativistic scattering involving two half spin fermions, $A$ and $B$, which are initially entangled with another fermion $C$ that does not participate in the scattering dynamics. We construct general expressions for the reduced spin matrices for the out-state considering a general tripartite spin-entangled state. In particular we study an inelastic QED process at tree-level, namely $e^-e^+\rightarrow \mu^- \mu^+$ and a half spin fermion $C$ as an spectator particle which can be entangled to the $AB$ system in the following ways: W state, GHZ state, $|\text{A}^\alpha \rangle \otimes |\Psi^{\pm} \rangle_{\text{BC}}$ and $|\text{A}^\alpha \rangle \otimes |\Phi^{\pm} \rangle_{\text{BC}}$, where $\{|\Psi^{\pm} \rangle,|\Phi^{\pm} \rangle\}$ are the Bell basis states and $|\text{A}^\alpha \rangle$ is a spin superposition state of system $A$. We calculate the von-Neumann entropy variation before and after the scattering for the particle $C$ and show that spin measurements in $C$ contain numerical information about the total cross section of the process. We compare the initial states W and GHZ as well as study the role played by the parameter $\alpha$ in the evaluation of the entropy variations and the cross section encoded in the spectator particle.