cs.thefarshad
hard

Quantum Teleportation

Move an unknown quantum state to a distant qubit using a shared Bell pair and two classical bits.

Quantum teleportation moves the state of one qubit to another, possibly far away, without ever sending the qubit itself. It is not science-fiction matter-transport — what travels is information, carried partly by entanglement and partly by two ordinary classical bits.

Alice holds an unknown state ∣ψ⟩=α∣0⟩+β∣1⟩|\psi\rangle = \alpha|0\rangle + \beta|1\rangle. She cannot just copy it (the no-cloning theorem forbids that) and she cannot measure it without destroying it. Step through the protocol below to watch ∣ψ⟩|\psi\rangle leave Alice and reappear, intact, on Bob’s qubit.

Alice2 qubits
|ψ⟩ = 0.866|0⟩ + 0.500|1⟩
Bob1 qubit
—
step 1/5: Alice has an unknown qubit

Alice holds |ψ⟩ = α|0⟩ + β|1⟩. She does not know α or β — and measuring would destroy it. The goal: recreate |ψ⟩ on Bob’s far-away qubit without sending the qubit itself.

The protocol

The four moving parts are entanglement, a joint measurement, a classical message, and a correction.

  1. Share a Bell pair. Ahead of time, Alice and Bob each take one half of an entangled pair 12(∣00⟩+∣11⟩)\frac{1}{\sqrt2}(|00\rangle + |11\rangle). This is the resource teleportation consumes.
  2. Bell measurement. Alice applies a CNOT (her unknown qubit → her Bell half) followed by a Hadamard, then measures both of her qubits. The outcome is two random classical bits m0m1m_0 m_1.
  3. Send two classical bits. Alice transmits m0m_0 and m1m_1 to Bob over a normal (classical) channel.
  4. Correct. Bob applies a gate determined by the bits: a ZZ if the first bit is 11, then an XX if the second is 11. His qubit becomes exactly ∣ψ⟩|\psi\rangle.

After Alice’s measurement, Bob’s qubit is in one of four states — the right ∣ψ⟩|\psi\rangle up to a known XX and/or ZZ. The classical bits tell him which of the four happened, so his correction always lands on the original state.

Why it does not break physics

Two worries come up immediately, and both resolve cleanly:

  • No faster-than-light signaling. Before Bob receives the classical bits, his qubit looks completely random — he learns nothing. The protocol is bottlenecked by the classical message, which obeys the speed of light.
  • No cloning. Alice’s measurement destroys her copy of ∣ψ⟩|\psi\rangle. At the end there is exactly one copy, now on Bob’s side — never two at once.

Why it matters

Teleportation is a foundational primitive, not a party trick. It is the way states are shuttled between modules in proposed quantum networks, the backbone of entanglement swapping in quantum repeaters, and a building block for fault-tolerant gates. It crisply demonstrates the division of labor in quantum information: entanglement plus classical communication can accomplish what neither can alone.

Takeaways

  • Teleportation transfers an unknown state using a shared Bell pair and 2 classical bits — the qubit itself never travels.
  • Alice’s Bell measurement plus Bob’s XX/ZZ correction reconstruct ∣ψ⟩|\psi\rangle exactly.
  • It respects no-cloning (Alice’s copy is destroyed) and cannot signal faster than light (the classical bits are required).

References