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Quantum Teleportation

intermediate3 qubits~12 min

Transfer a quantum state using entanglement and 2 classical bits

The question

How can you transfer an unknown quantum state to a distant qubit without physically sending the qubit, and without knowing what the state is?

Why this matters

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.

Intuition

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 $∣ ψ ⟩$ .

Key insight

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).

Circuit

Step-by-step walkthrough

Share a Bell pair between Alice and Bob

Prepare qubits 1 and 2 in $∣ Φ^{+} ⟩ = (∣00 ⟩ + ∣11 ⟩) / 2$ using H and CNOT. Alice keeps qubit 1, Bob keeps qubit 2.

∣ ψ ⟩_{0} \otimes \frac{∣00 ⟩ + ∣11 ⟩}{2}

Alice applies CNOT(q0, q1)

Alice entangles her unknown qubit (q0) with her half of the Bell pair (q1) using CNOT.

Alice applies H to q0

Hadamard on Alice’s original qubit completes the Bell-basis measurement preparation.

Alice measures q0 and q1

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.

Bob applies corrections

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.

∣ ψ ⟩ = α ∣0 ⟩ + β ∣1 ⟩ on Bob’s qubit

What the measurement tells you

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 $∣ ψ ⟩$ .

Full technical statement

Let $∣ ψ ⟩ = α$ $∣0 ⟩ + β$ $∣1 ⟩$ be the state to teleport. Qubits 1 and 2 share the Bell state $∣ Φ^{+} ⟩$ . The key mathematical step is rewriting the three-qubit joint state in the Bell basis for qubits 0 and 1. When you do this, you find that the joint state splits into four terms, each pairing a Bell state on qubits 0-1 with a specific transformation of $∣ ψ ⟩$ on qubit 2. Specifically: one term has the identity (no correction needed), one has Z applied, one has X applied, and one has XZ applied. Alice's Bell measurement on qubits 0-1 collapses the state to one of these four cases. Bob then applies the matching Pauli gate to recover $∣ ψ ⟩$ exactly.

Classical approach

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).

Quantum approach

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.

Tempting but wrong

It is tempting to think teleportation sends information faster than light. That is not what happens. Alice’s two measurement bits must be sent to Bob through an ordinary classical channel, which is limited by the speed of light. Until Bob receives those bits, his qubit looks completely random. The entanglement enables the transfer, but classical communication is still required to complete it.

Where this leads

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.

Gates you should know first

H gate →CNOT gate →X gate →Z gate →

Test your understanding

Why does quantum teleportation not violate the no-cloning theorem?

↗ Nielsen and Chuang, Quantum Computation and Quantum Information ↗ MIT OCW 8.06: entanglement notes

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Part of Quantum Algorithms