Genuine multipartite entanglement in the cluster-Ising model

We evaluate and analyze the exact value of a measure for local genuine tripartite entanglement in the one-dimensional cluster-Ising model for spin- $\frac{1}{2}$ particles. This model is attractive since cluster states are considered to be relevant sources for applying quantum algorithms and the model is experimentally feasible. Whereas bipartite entanglement is identically vanishing, we find that genuine tripartite entanglement is non zero in the anti-ferromagnetic phase and also in the cluster phase well before the critical point. We prove that the measure of local genuine tripartite entanglement captures all the properties of the symmetry-protected topological quantum phase transition. Remarkably, we find that the amount of genuine tripartite entanglement is independent of whether the ground states satisfy or break the symmetries of the Hamiltonian. We provide also strong evidences that local genuine tripartite entanglement represents the unique non-vanishing genuine multipartite entanglement.