Variation of entanglement entropy and mutual information in fermion-fermion scattering

We study the behavior of entanglement between different degrees of freedom of scattering fermions, based on an exemplary QED scattering process $e^{+}{e}^{-}\to {\mu }^{+}{\mu }^{-}$. The variation of entanglement entropy between two fermions from an initial state to the final state was computed, with respect to different entanglement between the ingoing particles. This variation of entanglement entropy is found to be proportional to an area quantity, the total cross section. We also study the spin-momentum and helicity-momentum entanglements within one particle in the aforementioned scattering process. The calculations of the relevant variations of mutual information in the same inertial frame reveals that, for a maximally entangled initial state, the scattering between the particles does not affect the degree of both of these entanglements of one particle in the final state. It is also found that the increasing degree of entanglement between two ingoing particles would restrict the generation of entanglement between spin (helicity) and momentum of one outgoing particle. And the entanglement between spin and momentum within one particle in the final state is shown to always be stronger than that for helicity-momentum for a general initial entanglement state, implying significantly distinct properties of entanglement for the helicity and spin perceived by an inertial observer.

Figure 1

The QED annihilation process $e^{+}{e}^{-}\to {\mu }^{+}{\mu }^{-}$ viewed in the center of mass frame and the corresponding lowest order Feynman diagram.

Figure 2

The variation of entanglement entropy $▵S_E$ as a function of the entanglement parameter $\eta$ of the initial state. $▵S_E$ is proportional to the bottom curve $g_1$ plus the top curve $f_1$ multiplied by $\log [\frac{V^2}{T^2}\frac{16E^4}{{\lambda }^2}]$. The left figure is for $\beta =0$. The right figure is for $\beta =\pi$.

Figure 3

The variation of mutual information $▵I(p_A,s_A)$ as a function of entanglement parameter $\eta .▵I(p_A,s_A)$ is proportional to function $g_2(\eta )$. The left figure is for the initial state with parameter $\beta =0$. The right figure is for the initial state with parameter $\beta =\pi$.

Figure 4

For the scattering process $e^{+}{e}^{-}\to {\mu }^{+}{\mu }^{-}$, the variation of mutual information $▵I$ in helicity representation (red curve) vs spin representation (blue curve) with the same inertial reference frame as a function of entanglement parameter $\eta$ form 0 to $\pi /2$. The left figure is for the initial state with parameter $\beta =0$. The right figure is for the initial state with parameter $\beta =\pi$.