  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.
