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Journal of Physics A Mathematical and Theoretical

Quantum entanglement in (d−1)-spherium

I. V. Toranzo, Angel Ricardo Plastino, Pablo Sánchez-Moreno, J. S. Dehesa

Published Oct 27, 2015• 9 citations
Publisher

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 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.

DOI
10.1088/1751-8113/48/47/475302

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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 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.
Quantum Entanglement in (d−1)(d-1)(d−1)-Spherium
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 sss-states of \emph{spherium}. This is a system of two particles (electrons) interacting via a Coulomb potential and confined to a (d−1)(d-1)(d−1)-sphere (that is, to the surface of a ddd-dimensional ball). We investigate the dependence of entanglement on the radius RRR of the system, on the spatial dimensionality ddd, and on energy. We find that entanglement increases monotonically with RRR, decreases with ddd, 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.
We explore the main entanglement properties exhibited by the eigenfunctions of two exactly soluble two-electron models, the Crandall atom and the Hooke atom, and compare them with the entanglement features of helium-like systems. We compute the amount of entanglement associated with the wavefunctions corresponding to the fundamental and first few excited states of these models. We investigate the dependence of the entanglement on the parameters of the models and on the quantum numbers of the eigenstates. It is found that the amount of entanglement of the system tends to increase with energy in both models. In addition, we study the entanglement of a few states of helium-like systems, which we compute using high-quality Kinoshita-like eigenfunctions. The dependence of the entanglement of helium-like atoms on the nuclear charge and on energy is found to be consistent with the trends observed in the previous two model systems.
The entanglement properties of two-electron atomic systems have been the subject of considerable research activity in recent years. These studies are still somewhat fragmentary, focusing on numerical computations on particular states of systems such as helium, or on analytical studies of model systems such as the Moshinsky atom. Some general trends are beginning to emerge from these studies: the amount of entanglement tends to increase with energy and, in the case of excited states, entanglement does not necessarily tend to zero in the limit of vanishing interaction between the two constituting particles. A physical explanation of these properties, shared by the different two-electron models investigated so far, is still lacking. As a first step towards this goal, we perform here, via a perturbative approach, an analysis of entanglement in two-electron models that sheds new light on the physical origin of the aforementioned features and on their universal character.
We present two methods of calculating the spatial entanglement of an interacting electron system within the framework of density-functional theory. These methods are tested on the model system of Hooke's atom for which the spatial entanglement can be exactly calculated. We analyze how the strength of the confining potential affects the spatial entanglement and how accurately the methods that we introduced reproduce the exact trends. We also compare the results to the outcomes of standard first-order perturbation methods. The accuracies of energies and densities when using these methods are also considered.