  In this paper we study the increment of the entanglement entropy and of the
(replica) logarithmic negativity in a zero-density excited state of a free
massive bosonic theory, compared to the ground state. This extends the work of
two previous publications by the same authors. We consider the case of two
disconnected regions and find that the change in the entanglement entropy
depends only on the combined size of the regions and is independent of their
connectivity. We subsequently generalize this result to any number of
disconnected regions. For the replica negativity we find that its increment is
a polynomial with integer coefficients depending only on the sizes of the two
regions. The logarithmic negativity turns out to have a more complicated
functional structure than its replica version, typically involving roots of
polynomials on the sizes of the regions. We obtain our results by two methods
already employed in previous work: from a qubit picture and by computing
four-point functions of branch point twist fields in finite volume. We test our
results against numerical simulations on a harmonic chain and find excellent
agreement.
