The entanglement of few-particle systems when using the local-density approximation

TL;DR

This paper explores calculating entanglement in many-particle systems using density-functional theory and the local-density approximation, introducing an interacting LDA system to compare with exact entanglement in model atoms.

Contribution

It introduces the concept of an interacting LDA system to evaluate entanglement, providing a new approach to assess the local-density approximation's accuracy in quantum information contexts.

Findings

Interacting LDA system accurately reproduces ground state densities.

Calculated spatial entanglement closely matches exact values for test systems.

Position-space entropy may serve as a proxy for entanglement.

Abstract

In this chapter we discuss methods to calculate the entanglement of a system using density-functional theory. We firstly introduce density-functional theory and the local-density approximation (LDA). We then discuss the concept of the `interacting LDA system'. This is characterised by an interacting many-body Hamiltonian which reproduces, uniquely and exactly, the ground state density obtained from the single-particle Kohn-Sham equations of density-functional theory when the local-density approximation is used. We motivate why this idea can be useful for appraising the local-density approximation in many-body physics particularly with regards to entanglement and related quantum information applications. Using an iterative scheme, we find the Hamiltonian characterising the interacting LDA system in relation to the test systems of Hooke's atom and helium-like atoms. The interacting LDA system ground state wavefunction is then used to calculate the spatial entanglement and the results are compared and contrasted with the exact entanglement for the two test systems. For Hooke's atom we also compare the entanglement to our previous estimates of an LDA entanglement. These were obtained using a combination of evolutionary algorithm and gradient descent, and using an LDA-based perturbative approach. We finally discuss if the position-space information entropy of the density---which can be obtained directly from the system density and hence easily from density-functional theory methods---can be considered as a proxy measure for the spatial entanglement for the test systems.