  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.
