Quantum Teleportation

qc/entanglement qc/gates

Transfers an unknown quantum state from Alice to Bob using a pre-shared Bell pair plus a classical channel — not matter transport, despite the name. Alice ends up with nothing (her copy is destroyed by measurement, consistent with the No-Cloning Theorem); Bob ends up with the exact original state.

Setup

Alice holds two qubits: Q (the unknown state to send) and A (her half of a pre-shared Bell pair). Bob holds B, the other half of that same Bell pair.

The protocol

1. Entangle and measure. Alice applies CX(Q, A) then H(Q), then measures both qubits: $Q\to c_1$, $A\to c_0$.
2. Classical channel. Alice sends the two classical bits $(c_0,c_1)$ to Bob — over an ordinary, light-speed-limited classical channel.
3. Conditional correction. Bob applies X on B if $c_1=1$, then Z on B if $c_0=1$. B now holds exactly the state Q started in.

from qiskit import QuantumCircuit
 
qc = QuantumCircuit(3, 3)
# ... prepare qc's qubit 0 (Q) in the unknown state to send, qubits 1(A)/2(B) as a Bell pair ...
qc.cx(0, 1)
qc.h(0)
qc.measure([0, 1], [1, 0])          # c1 <- Q, c0 <- A
with qc.if_test((qc.clbits[1], 1)):  # if c1
    qc.x(2)
with qc.if_test((qc.clbits[0], 1)):  # if c0
    qc.z(2)

Why this doesn’t break relativity

Measuring A instantly affects the joint state, but Bob’s qubit is just noise until he applies the correction — and he can’t know which correction to apply without the two classical bits, which travel no faster than light. The entanglement carries correlation, not information; the classical channel is what actually carries the information (see CHSH Inequality and Bell Tests for the same “correlation without signaling” point in a different context).

Key insight: entanglement is consumed

The Bell pair is used up by this protocol — it can’t be reused for a second teleportation. Entanglement here is a resource, spent to buy the transfer of one qubit’s worth of quantum information using only two classical bits.

First proposed in 1993; demonstrated over 1,400 km lab-to-satellite. Considered the prototype building block for quantum networking and quantum repeaters.

Related

• Bell States
• No-Cloning Theorem — why Alice’s original has to be destroyed, not copied
• CHSH Inequality and Bell Tests
• Dynamic Circuits — the same mid-circuit-measurement + classical-feedforward pattern

Self-Check

• Could you walk through all three steps of the teleportation protocol from memory?
• Why does teleportation need a classical channel at all — what would go wrong without it?
• What happens to Alice’s copy of the state, and why does that matter for whether this violates no-cloning?
• Why must Bob’s correction gates be classically conditioned on Alice’s measurement outcomes rather than applied unconditionally?
• If you look at the raw measurement counts of a simulated teleportation circuit including Alice’s classical bits, why do they look wrong until you marginalize those bits out and keep only Bob’s?