Generalized Teleportation and Entanglement Recycling

We introduce new teleportation protocols which are generalizations of the original teleportation protocols that use the Pauli group and the port-based teleportation protocols, introduced by Hiroshima and Ishizaka, that use the symmetric permutation group. We derive sufficient conditions for a set of operations, which in general need not form a group, to give rise to a teleportation protocol and provide examples of such schemes. This generalization leads to protocols with novel properties and is needed to push forward new schemes of computation based on them. Port-based teleportation protocols and our generalizations use a large resource state consisting of $N$ singlets to teleport only a single qubit state reliably. We provide two distinct protocols which recycle the resource state to teleport multiple states with error linearly increasing with their number. The first protocol consists of sequentially teleporting qubit states, and the second teleports them in a bulk.