Quantum Entanglement in $(d-1)$-Spherium

TL;DR

This paper analytically calculates the entanglement of eigenstates in a two-electron system confined to a $(d-1)$-sphere, revealing how entanglement varies with system size, dimension, and energy.

Contribution

It provides the first exact analytical expressions for entanglement in spherium, a model of two Coulomb-interacting particles on a sphere, and analyzes its dependence on key parameters.

Findings

Entanglement increases with the sphere radius R.

Entanglement decreases as the spatial dimension d increases.

Entanglement tends to increase with the energy of eigenstates.

Abstract

There are very few systems of interacting particles (with continuous variables) for which the entanglement of the concomitant eigenfunctions can be computed in an exact, analytical way. Here we present analytical calculations of the amount of entanglement exhibited by $s$-states of \emph{spherium}. This is a system of two particles (electrons) interacting via a Coulomb potential and confined to a $(d-1)$-sphere (that is, to the surface of a $d$-dimensional ball). We investigate the dependence of entanglement on the radius $R$ of the system, on the spatial dimensionality $d$, and on energy. We find that entanglement increases monotonically with $R$, decreases with $d$, and also tends to increase with the energy of the eigenstates. These trends are discussed and compared with those observed in other two-electron atomic-like models where entanglement has been investigated.