Transfer a quantum state using entanglement and 2 classical bits
How can you transfer an unknown quantum state to a distant qubit without physically sending the qubit, and without knowing what the state is?
Quantum teleportation, proposed by Bennett et al. in 1993, is one of the most remarkable protocols in quantum information. It transfers an unknown quantum state from one location to another using a shared Bell pair and two classical bits of communication. No physical qubit travels — only classical information. Teleportation does not violate no-cloning (the original state is destroyed) or faster-than-light communication (classical bits must be sent). It is foundational to quantum networks, distributed quantum computing, and fault-tolerant quantum computing via gate teleportation.
Alice has a qubit in an unknown state |ψ⟩ = α|0⟩ + β|1⟩ that she wants to send to Bob. They share a Bell pair. Alice performs a Bell-basis measurement on her unknown qubit and her half of the Bell pair. This measurement yields 2 classical bits and collapses Bob’s qubit into one of four states, each related to |ψ⟩ by a known Pauli operation. Alice sends her 2 bits to Bob, who applies the corresponding correction (I, X, Z, or XZ). Bob’s qubit is now in state |ψ⟩.
Bell measurement on Alice’s side teleports the state information onto Bob’s qubit via the pre-shared entanglement. The classical bits tell Bob which Pauli correction to apply. The original state is destroyed in the process (no-cloning is respected).
Prepare qubits 1 and 2 in |Φ⁺⟩ = (|00⟩+|11⟩)/√2 using H and CNOT. Alice keeps qubit 1, Bob keeps qubit 2.
Alice entangles her unknown qubit (q0) with her half of the Bell pair (q1) using CNOT.
Hadamard on Alice’s original qubit completes the Bell-basis measurement preparation.
Alice measures both her qubits, obtaining two classical bits (b0, b1). This collapses Bob’s qubit into a state related to |ψ⟩ by a known Pauli operation.
Based on Alice’s bits: if b1=1, Bob applies X. If b0=1, Bob applies Z. After correction, Bob’s qubit is in state |ψ⟩ — teleportation complete.
Alice’s two measurement bits (b0, b1) determine which Pauli correction Bob needs: (0,0)→I, (0,1)→X, (1,0)→Z, (1,1)→XZ. After correction, Bob has the original state |ψ⟩.
Let |ψ⟩ = α|0⟩ + β|1⟩ be the state to teleport, and let qubits 1,2 share |Φ⁺⟩. The joint state of all three qubits can be rewritten as: |ψ⟩₂₁₂ = ½[|Φ⁺⟩₀₁(α|0⟩+β|1⟩)₂ + |Φ⁻⟩₀₁(Z(α|0⟩+β|1⟩))₂ + |Ψ⁺⟩₀₁(X(α|0⟩+β|1⟩))₂ + |Ψ⁻⟩₀₁(XZ(α|0⟩+β|1⟩))₂]. Bell measurement on qubits 0,1 collapses qubit 2 into one of these four cases, each correctable by a Pauli gate.
Classically, to send a continuous state you’d need infinite precision — infinite classical bits. You also can’t copy it without knowing it (no-cloning).
Teleportation transfers the full quantum state using only 1 pre-shared Bell pair and 2 classical bits, regardless of the state’s complexity. The original is destroyed, respecting no-cloning.
Quantum teleportation does not transmit information faster than light. Alice’s measurement results must be sent to Bob via a classical channel (limited by the speed of light) before he can apply the correct Pauli gate. Without Alice’s bits, Bob’s qubit looks completely random.
Teleportation generalizes to gate teleportation (teleporting a gate operation), which is central to fault-tolerant quantum computing. It also underpins quantum repeaters for long-distance quantum communication.
Why does quantum teleportation not violate the no-cloning theorem?