  We quantify the extractable entanglement of excited states of a Lieb-Liniger
gas that are obtained from coarse-grained measurements on the ground state in
which the boson number in one of two complementary contiguous partitions of the
gas is determined. Numerically exact results obtained from the coordinate Bethe
ansatz show that the von Neumann entropy of the resulting bipartite pure state
increases monotonically with the strength of repulsive interactions and
saturates to the impenetrable boson limiting value. We also present evidence
indicating that the largest amount of entanglement can be extracted from the
most probable projected state having half the number of bosons in a given
partition. Our study points to a fundamental difference between the nature of
the entanglement in free-bosonic and free-fermionic systems, with the
entanglement in the former being zero after projection, while that in the
latter (corresponding to the impenetrable boson limit) being non-zero.
