Teleportation as a Channel

Quantum teleportation implements the identity channel on an unknown state using shared entanglement and two classical bits.

Moving a state without moving the system

Teleportation transfers an unknown qubit state from Alice to Bob without sending the qubit itself. They pre-share a Bell pair. Alice performs a joint Bell measurement on her unknown qubit and her half of the pair, obtaining two classical bits. She sends those bits to Bob, who applies a corresponding Pauli correction to his half. His qubit then carries the original state exactly.

Viewed as a channel

End to end, teleportation realizes the identity channel: whatever state entered Alice's side emerges on Bob's side. With a perfect Bell pair and correct corrections, the effective map is E(rho) = rho. Imperfections turn it into a noisy channel; teleporting through a partially entangled or noisy pair yields a channel whose fidelity is set by the quality of the shared entanglement, making teleportation a way to convert entanglement quality into channel quality.

import numpy as np
# the four corrections are the Paulis, one per 2-bit outcome
I=np.eye(2);X=np.array([[0,1],[1,0]])
Z=np.array([[1,0],[0,-1]]);Y=1j*X@Z
print('corrections:', 'I,X,Z,ZX for 00,01,10,11')

No cloning, no signaling

Teleportation destroys the original during the Bell measurement, respecting the no-cloning theorem: no second copy ever exists. It also respects no-signaling: before Bob receives the two classical bits, his qubit is maximally mixed and carries nothing. The classical message, limited to light speed, is what makes the protocol causal. Entanglement alone transmits no information.

Why it matters

Teleportation is the routing primitive of quantum networks and repeaters: entanglement is distributed and purified, then teleportation moves quantum data across it. In fault-tolerant computing, gate teleportation injects hard-to-implement gates, such as the T gate, by teleporting through specially prepared resource states. Teleportation as a channel is therefore both a communication tool and a computational building block.