  We investigate entanglement properties of the excited states of the spin-1/2
Heisenberg (XXX) chain with isotropic antiferromagnetic interactions, by
exploiting the Bethe ansatz solution of the model. We consider eigenstates
obtained from both real and complex solutions ("strings") of the Bethe
equations. Physically, the former are states of interacting magnons, whereas
the latter contain bound states of groups of particles. We first focus on the
situation with few particles in the chain. Using exact results and
semiclassical arguments, we derive an upper bound S_MAX for the entanglement
entropy. This exhibits an intermediate behavior between logarithmic and
extensive, and it is saturated for highly-entangled states. As a function of
the eigenstate energy, the entanglement entropy is organized in bands. Their
number depends on the number of blocks of contiguous Bethe-Takahashi quantum
numbers. In presence of bound states a significant reduction in the
entanglement entropy occurs, reflecting that a group of bound particles behaves
effectively as a single particle. Interestingly, the associated entanglement
spectrum shows edge-related levels. At finite particle density, the
semiclassical bound S_MAX becomes inaccurate. For highly-entangled states
S_A\propto L_c, with L_c the chord length, signaling the crossover to extensive
entanglement. Finally, we consider eigenstates containing a single pair of
bound particles. No significant entanglement reduction occurs, in contrast with
the few-particle case.
