Measuring QED cross sections via entanglement

We considered a QED scattering ($AB\ensuremath{\rightarrow}AB$), 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 was evaluated and we noted coherence (off-diagonal) terms were created, which led to non-null values for $⟨{\ensuremath{\sigma}}_{x}⟩$ and $⟨{\ensuremath{\sigma}}_{y}⟩$ that are, in principle, measurable in a Stern-Gerlach apparatus. We chose a particular QED scattering (${e}^{+}{e}^{\ensuremath{-}}\ensuremath{\rightarrow}{\ensuremath{\mu}}^{+}{\ensuremath{\mu}}^{\ensuremath{-}}$) and found that $⟨{\ensuremath{\sigma}}_{x}⟩$ and $⟨{\ensuremath{\sigma}}_{y}⟩$ are proportional to the total cross section (${\ensuremath{\sigma}}_{\text{total}}$) 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 information ${I}_{AC}$ and ${I}_{BC}$, and noticed an increase (decrease) in ${I}_{AC}$ (${I}_{BC}$), which indicates that, after $AB$ interact, the total amount of correlations ($\mathrm{quantum}+\mathrm{classical}$) is distributed among the 3 subsystems.