You can't copy an unknown qubit, and measuring destroys it — yet its exact state can be moved across the world using one entangled pair and two classical bits. Here is the whole trick, step by step and in symbols.
Teleportation is a fax machine that shreds the original. You feed in a page; the machine reads it — destroying it in the act — sends a short code down the phone line, and an identical page prints at the far end. No paper ever crossed the wire, only instructions. And because the original was shredded, there's never two copies in the world at once.
Every quantum algorithm is drawn the same way: horizontal wires are qubits, time flows left to right, and boxes are gates. Here is teleportation in that language — three wires, four gates, two measurements, and a classically-controlled fix on Bob's qubit.
The dashed lines are classical wires — ordinary bits, not qubits. They carry Alice's two measurement results to Bob, where they pick which of four corrections to apply. No quantum information ever travels along them.
It looks like magic until you watch the algebra rearrange itself. Start with the unknown qubit beside the shared Bell pair (Alice's two qubits written first, Bob's last):
Apply Alice's CNOT then her Hadamard, and the whole thing regroups into four equal pieces — each tagged by the two bits she is about to read:
Look at the trailing factor in each line — that is Bob's qubit. The instant Alice measures and gets some ab, Bob's qubit is left in exactly one of those four expressions — each a simple gate away from the original α|0⟩ + β|1⟩. That is why the corrections below are all he needs.
Alice's joint measurement randomly lands in one of four equally-likely outcomes. Each leaves Bob's qubit holding ψ up to a known Pauli error — and the two bits tell him exactly which one to undo:
Two facts keep the universe honest. No cloning: ψ is destroyed at Alice the instant she measures, so only one copy ever exists. No faster-than-light: until the classical bits arrive, Bob's qubit is an even mixture — useless. The state moved, but it rode on an ordinary phone call.
Not a transporter. No matter or energy moves — only a state, a list of amplitudes. Bob already had a qubit; teleportation just reconfigures it to match ψ.
Not a copy. The original is destroyed at Alice's measurement, so there is never a second ψ in existence — the no-cloning theorem stays intact.
Not faster than light. Without Alice's two classical bits, Bob's qubit is indistinguishable from noise. The protocol runs no faster than that phone call.