Measuring QED cross sections via entanglement

We considered 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 was evaluated and we noted coherence (off-diagonal) terms were created, which led 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 $⟨{\sigma }_x⟩$ and $⟨{\sigma }_y⟩$ 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 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.

Figure 1

QED scattering with a witness particle. The particles $B$ and $C$ are initially entangled in spin. After the scattering, the three particles become entangled.

Figure 2

$e^{+}{e}^{-}\to {\mu }^{+}{\mu }^{-}$ scattering with a witness particle $C$. The quantity $\mathrm{\Lambda }$ is found to be proportional to cross section if the $A$ beam is unpolarized.

Figure 3

Plot of ${(I_{AC})}_f$ for the set of parameters ($T$, $V$, $E_{{\mathit{p}}_i}$, $e\to 1$; $\mathrm{\Lambda }\to e^4/3\pi$). The angle $\eta$ was split in $n=1000$ parts from 0 to $\pi /2$ in order to run the plot. The final mutual information is maximal for $\eta =\pi /4$—compatible with a Bell basis for $BC$’s initial state. Conversely, for $\eta =0$ or $\eta =\pi /2$, i.e., $B$ and $C$ initially unentangled, there is no mutual information between $A$ and $C$.

Figure 4

Initial (dashed line) and final (thick line) mutual information between $B$ and $C$. There is a decrease, which is largest for $\eta =\pi /4$, for part of the correlations are transferred to the partition $AC$. The plot was made using the set of parameters ($T$, $V$, $E_{{\mathit{p}}_i}$, $e\to 1$; $\mathrm{\Lambda }\to e^4/3\pi$). The angle $\eta$ was split in $n=1000$ parts from 0 to $\pi /2$ in order to run the plot.