Entanglement and Entropy in Electron-Electron Scattering

TL;DR

This paper investigates how Coulomb scattering of two electrons creates momentum and spin entanglement, estimating the von Neumann entropy from experimental Shannon entropy, and analyzing the effects of energy, spin, and post-selection on entanglement.

Contribution

It introduces a discretisation method to estimate von Neumann entropy from Shannon entropy in electron scattering and analyzes entanglement dependence on energy and spin configurations.

Findings

Entropy is high at low energies, indicating strong momentum entanglement.

Entropy drops near zero at around 10 keV when azimuthal degrees are integrated out.

Post-selection at scattering angle pi/2 yields strong spin entanglement.

Abstract

Treating Coulomb scattering of two free electrons in a stationary approach, we explore the momentum and spin entanglement created by the interaction. We show that a particular discretisation provides an estimate of the von Neumann entropy of the one-electron reduced density matrix from the experimentally accessible Shannon entropy. For spinless distinguishable electrons the entropy is sizeable at low energies, indicating strong momentum entanglement, and drops to almost zero at energies of the order of 10 keV when the azimutal degree of freedom is integrated out, i.e. practically no entanglement and almost pure one-electron states. If spin is taken into account, the entropy for electrons with antiparallel spins should be larger than in the parallel-spin case, since it embodies both momentum and spin entanglement. Surprisingly, this difference, as well as the deviation from the spin-less case, is extremely small for the complete scattering state. Strong spin entanglement can however be obtained by post-selecting states at scattering angle pi/2.