Quantum teleportation
Moving a qubit's state to another qubit elsewhere — without sending the qubit, and without exceeding the speed of light.
The protocol was described in 1993 by Bennett and colleagues. The parties share an entangled pair in advance. The sender measures her qubit together with the one she wants to send, gets two classical bits as the result, and transmits them by ordinary means. The receiver uses them to pick the right correction, and his qubit then lands in the original state.
Nothing material moves, and the two classical bits are necessary — so nothing happens faster than light. The original is also destroyed in the measurement, entirely in keeping with the no-cloning theorem. The technique is not science fiction but laboratory routine, and underpins moving states between parts of a quantum computer.