Measuring a QED cross section via a witness particle

We consider a QED scattering ($AB\to AB$), in which $B$ is initially entangled with a third particle ($C$) that does not participate directly in the scattering. The effect of the scattering over $C$’s final state is evaluated and we note coherence (off-diagonal) terms are created, which lead to non null values for $⟨{\sigma }_x⟩$ and $⟨{\sigma }_y⟩$ that are, in principle, measurable in a Stern-Gerlach apparatus. We chose a particular QED scattering ($e^{+}{e}^{-}\to {\mu }^{+}{\mu }^{-}$) and found that $\langle \sigma_x\rangle$and$\langle \sigma_y\rangle$ are proportional to the total cross section (${\sigma }_{\text{total}}$) of the $AB$ scattering, besides being maximal if $BC$’s initial state is taken as a Bell basis. Furthermore, we calculated the initial and final mutual informations $I_{AC}$ and $I_{BC}$, and noticed an increase (decrease) in $I_{AC}$ ($I_{BC}$), which indicates that, after $AB$ interact, the total amount of correlations (quantum $+$ classical) are distributed among the $3$ subsystems.