Entanglement and the Mott insulator--superfluid phase transition in bosonic atom chains.

We analyze the developing of bipartite and multipartite entanglement through the Mott insulator-superfluid quantum phase transition. Starting from a Mott insulator state, where a filling factor ν = N/M = 1 per lattice site is considered, we derive an expression for a completely connected graph configuration of bosons and show how bipartite and multipartite entanglement evolve through the phase transition predicted in previous works. We show how, in the transition, bipartite entanglement is distributed through the system respecting monogamous relations.