  Entanglement is an essential property of quantum many-body systems. However,
its local detection is challenging and was so far limited to spin degrees of
freedom in ion chains. Here we measure entanglement between the spins of atoms
located on two lattice sites in a one-dimensional Bose-Hubbard chain which
features both local spin- and particle-number fluctuations. Starting with an
initially localized spin impurity, we observe an outwards propagating
entanglement wave and show quantitatively how entanglement in the spin sector
rapidly decreases with increasing particle-number fluctuations in the chain.
