  We compute entanglement entropy (EE) of a spherical region in
$(3+1)$-dimensional $\mathcal{N}=4$ supersymmetric $SU(N)$ Yang-Mills theory in
states described holographically by probe D3-branes in $AdS_5 \times S^5$. We
do so by generalising methods for computing EE from a probe brane action
without having to determine the probe's back-reaction. On the Coulomb branch
with $SU(N)$ broken to $SU(N-1)\times U(1)$, we find the EE monotonically
decreases as the sphere's radius increases, consistent with the $a$-theorem.
The EE of a symmetric-representation Wilson line screened in $SU(N-1)$ also
monotonically decreases, although no known physical principle requires this. A
spherical soliton separating $SU(N)$ inside from $SU(N-1)\times U(1)$ outside
had been proposed to model an extremal black hole. However, we find the EE of a
sphere at the soliton's radius does not scale with the surface area. For both
the screened Wilson line and soliton, the EE at large radius is described by a
position-dependent W-boson mass as a short-distance cutoff. Our holographic
results for EE and one-point functions of the Lagrangian and stress-energy
tensor show that at large distance the soliton looks like a Wilson line in a
direct product of fundamental representations.
