Concatenated quantum teleportation

Effective quantum teleportation must be able to teleport states or parts of states in arbitrarily large Hilbert spaces, which will require concatenation of quantum teleportation processes that operate for fixed dimension. We compute the exact effective fidelity for conatenated quantum teleportation and analyze special cases to identify the minimum performance thresholds for each teleportation channel that is required to claim that the entire teleportation is \'quantum\'. We point out potential applications in sharing quantum secrets, open-destination teleportation, and distributing quantum information in quantum networks.