An Area Law for Entanglement Entropy in Particle Scattering

Abstract

The scattering cross section is the effective area of collision when two particles collide. Quantum mechanically, it is a measure of the probability for a specific process to take place. Employing wave packets to describe the scattering process, we compute the entanglement entropy in 2-to-2 scattering of particles in a general setting using the $S$-matrix formalism. Applying the optical theorem, we show that the linear entropy ${\mathcal{E}}_2$ is given by the elastic cross section ${\sigma }_{\text{el}}$ in unit of the transverse size $L^2$ of the wave packet, ${\mathcal{E}}_2\sim {\sigma }_{\text{el}}/L^2$, when the initial states are not entangled. The result allows for dual interpretations of the entanglement entropy as an area and as a probability. Since ${\sigma }_{\text{el}}$ is generally believed, and observed experimentally, to grow with the collision energy $\sqrt{s}$ in the high energy regime, the result suggests a "second law" of entanglement entropy for high energy collisions. Furthermore, the Froissart bound places an upper limit on the entropy growth.

Comments: 5 pages + Supplementary Material, 1 figure