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 probes 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 spheres 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 solitons 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.
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