Abstract
Teleportation allows Alice to send a pre-prepared quantum state to Bob using only pre-shared entanglement and classical communication. Here we show that it is possible to teleport a state which is also post-selected. Post-selection of a state Φ means that after Alice has finished her experiment she performs a measurement and only keeps runs of the experiment where the measurement outcome is Φ. We also demonstrate pre and post-selected port-based teleportation. Finally we use these protocols to perform instantaneous non-local quantum computation on pre and post-selected systems, and significantly reduce the entanglement required to instantaneously measure an arbitrary non-local variable of spatially separated pre and post-selected systems.