  We present a detailed study of the Entanglement Entropy (EE) of excited
states in all closed rank one subsectors of N=4 SYM, namely SU(2), SU(1|1) and
SL(2). Exploiting the techniques of the Coordinate and the Algebraic Bethe
Ansatz we obtain the EE for spin chains with up to seven magnons, at leading
order in the coupling expansion but exact in the length of the spin chain and
of the part of it that we cut. Focusing on the superconformal primary operator
with two magnons in the BMN limit, we derive analytic and exact, in the
coupling $\lambda'$, expressions for the Renyi and the EE. The interpolating
functions for the Renyi and the EE monotonically increase as the coupling
increases from the weak coupling $\lambda' \rightarrow 0$ regime to the strong
coupling $\lambda' \rightarrow \infty$ regime. This results to a violation of a
certain bound for the EE that is present at weak coupling and confirms the
physical intuition that entanglement increases when the coupling increases.
