Quantum Teleportation

What You'll Learn: How to transfer a quantum state from one qubit to another using a shared entangled pair and two classical bits — without physically moving the qubit.

Level: Beginner | Time: 20 minutes | Qubits: 3 | Framework: Qiskit

Prerequisites: Bell State

The Idea

Suppose Alice has a qubit in some unknown state |ψ⟩ and wants to send it to Bob. She can't just measure it and send the result — measurement destroys the quantum state. She also can't copy it (no-cloning theorem). So how can she transfer the information?

The answer: quantum teleportation. If Alice and Bob share a pre-existing entangled pair (a Bell state), Alice can perform a special measurement on her qubit and her half of the Bell pair, send the two classical bits of the measurement result to Bob, and Bob can use those bits to reconstruct the exact state |ψ⟩ on his qubit.

The original state is destroyed at Alice's end (it has to be — otherwise we'd have cloning). And Bob needs the classical bits before he can do anything — so there's no faster-than-light communication. But the fidelity is perfect: Bob gets exactly the state Alice had.

This protocol was proposed by Bennett, Brassard, Crépeau, Jozsa, Peres, and Wootters in 1993 and first demonstrated experimentally in 1997.

How It Works

The Circuit

CODECopy
         ┌─────────┐     ┌───┐
q_0: ────┤ prepare  ├──■──┤ H ├─── M ───────────────
         └─────────┘┌─┴─┐└───┘     │
q_1: ───────■───────┤ X ├───────── M ──────■────────
          ┌─┴─┐     └───┘                  │
q_2: ─────┤ X ├────────────────────────────X── Z ── M
          └───┘                       (conditioned)

Step 1: Share a Bell Pair

PYTHONCopy
qc.h(1)       # Superposition on qubit 1
qc.cx(1, 2)   # Entangle qubits 1 and 2: (|00⟩ + |11⟩)/√2

Alice gets qubit 1, Bob gets qubit 2. They can be anywhere in the universe — the entanglement persists regardless of distance.

Step 2: Alice's Bell Measurement

PYTHONCopy
qc.cx(0, 1)   # CNOT: entangle message qubit with Alice's Bell qubit
qc.h(0)       # Hadamard: complete the Bell basis measurement
qc.measure(0, 0)  # Measure → one of four outcomes (00, 01, 10, 11)
qc.measure(1, 1)

Alice now has two classical bits. This measurement "projects" the three-qubit system into a state where Bob's qubit is |ψ⟩ — up to a known correction that depends on Alice's bits.

Step 3: Bob's Corrections

PYTHONCopy
# If Alice's bit 1 = 1, apply X (bit flip)
with qc.if_test((cr[1], 1)):
    qc.x(2)
# If Alice's bit 0 = 1, apply Z (phase flip)
with qc.if_test((cr[0], 1)):
    qc.z(2)

After corrections, Bob's qubit is exactly in state |ψ⟩.

The Math

Setup

Let Alice's unknown state be |ψ⟩ = α|0⟩ + β|1⟩ and the shared Bell pair be |Φ+⟩ = (|00⟩ + |11⟩)/√2 on qubits 1, 2.

The initial 3-qubit state is:

|ψ⟩₀ ⊗ |Φ+⟩₁₂ = (α|0⟩ + β|1⟩) ⊗ (|00⟩ + |11⟩)/√2

After Bell Measurement

Rewriting in the Bell basis on qubits 0, 1:

= ½ [ |Φ+⟩₀₁(α|0⟩ + β|1⟩)₂ + |Φ-⟩₀₁(α|0⟩ − β|1⟩)₂ + |Ψ+⟩₀₁(α|1⟩ + β|0⟩)₂ + |Ψ-⟩₀₁(α|1⟩ − β|0⟩)₂ ]

Each Bell measurement outcome corresponds to Bob having a specific transformation of |ψ⟩. The correction gates (I, X, Z, XZ) undo that transformation.

Why It Works

The key insight is that the Bell measurement doesn't reveal α or β — it only tells Bob which correction to apply. The quantum information is never transmitted classically. It's "hidden" in the correlations of the entangled pair.

Expected Output

When teleporting the |+⟩ state with 1024 shots:

Bob's qubit (bit 2) is what matters: after correction, it should have the same statistics as the original state.

Running the Circuit

PYTHONCopy
from circuit import run_circuit, verify_teleportation

# Teleport the |+⟩ state (default)
counts = run_circuit(shots=1024)

# Teleport |1⟩
counts = run_circuit(shots=1024, initial_state="one")

# Verify all four cardinal states
result = verify_teleportation(shots=4096)
for check in result["checks"]:
    print(f"  {check['name']}: {check['detail']}")

Try It Yourself

1. Teleport |1⟩: Change initial_state="one" and verify Bob always gets 1.

2. Remove corrections: Comment out Bob's X and Z gates. What happens to the teleportation fidelity?

3. Verify no-cloning: After the protocol, measure Alice's qubit 0. Is it still in state |ψ⟩? (It shouldn't be — it collapsed during the Bell measurement.)

4. Entanglement swapping: Chain two teleportations to create entanglement between qubits that never interacted directly.

What's Next

• Superdense Coding — The "dual" protocol: send 2 classical bits using 1 qubit + entanglement
• GHZ State — Extend entanglement to 3+ qubits
• Bell State — Review the entangled pair that makes teleportation possible

Applications

References

• Bennett, C.H., Brassard, G., Crépeau, C., Jozsa, R., Peres, A., Wootters, W.K. (1993). "Teleporting an Unknown Quantum State via Dual Classical and Einstein-Podolsky-Rosen Channels." Physical Review Letters 70(13), 1895-1899. DOI: 10.1103/PhysRevLett.70.1895
• Bouwmeester, D. et al. (1997). "Experimental quantum teleportation." Nature 390, 575-579.
• Nielsen, M.A. & Chuang, I.L. Quantum Computation and Quantum Information, Section 1.3.7 (Quantum teleportation).