Equivalence between Entanglement and the Optimal Fidelity of Continuous Variable Teleportation

We devise the optimal form of Gaussian resource states enabling continuous variable teleportation with maximal fidelity. We show that a nonclassical optimal fidelity of $N$-user teleportation networks is {\it necessary and sufficient} for $N$-party entangled Gaussian resources, yielding an estimator of multipartite entanglement. This {\it entanglement of teleportation} is equivalent to entanglement of formation in the two-user protocol, and to localizable entanglement in the multi-user one. The continuous-variable tangle, quantifying entanglement sharing in three-mode Gaussian states, is operationally linked to the optimal fidelity of a tripartite teleportation network.