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 $\mathit{\text{et}}\mathit{\text{al.}}$, Phys. Rev. Lett. $\mathbf{\text{109}}$, 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.