Continuous-variable-entanglement swapping and its local certification: Entangling distant mechanical modes

We introduce a modification of the standard entanglement swapping protocol where the generation of entanglement between two distant modes is realized and verified using only local optical measurements. We show, indeed, that a simple condition on the purity of the initial state involving also an ancillary mode is sufficient to guarantee the success of the protocol by local measurements [M. Abdi, S. Pirandola, P. Tombesi, and D. Vitali, Phys. Rev. Lett. 109, 143601 (2012)]. We apply the proposed protocol to a tripartite optomechanical system where the never interacting mechanical modes become entangled and certified using only local optical measurements.