Teleportation

State transfer, not matter transfer

Teleportation moves the state of a qubit from Alice's hardware to Bob's without moving the qubit itself. Nothing material travels; nothing is copied; nothing arrives before the classical message does. The protocol consumes one shared Bell pair and two classical bits, and it destroys the original in the process.

The ledger, up front:

• input: one unknown qubit state at Alice's side;
• shared resource: one Bell pair, split between Alice and Bob;
• classical communication: two bits from Alice to Bob;
• output: Bob's qubit in the original state, after his correction;
• side effect: Alice's original is gone — her measurement consumed it.

The protocol, precisely

Write the unknown input as:

Alice and Bob share a Bell pair:

Alice entangles the input with her half of the pair — CNOT from the input to her half, then H on the input — and measures both of her qubits. The measurement update rule describes what remains:

Her two measurement bits identify which of four conditional states Bob's qubit now holds; each is $|\psi ⟩$ up to one known correction from $\{I,X,Z,XZ\}$. Alice sends the bits; Bob applies the matching correction; his qubit is exactly $|\psi ⟩$. Until the bits arrive, Bob's qubit is — to him — a maximally mixed object carrying nothing.

Worked example: the three-wire trace

Lay out the wires:

q0: unknown state |ψ>   (Alice)
q1: Alice's half of the Bell pair
q2: Bob's half of the Bell pair

The steps:

1. Prepare the Bell pair on q1 and q2: H on q1, then CNOT q1→q2.
2. CNOT q0→q1, then H on q0 — Alice's Bell-basis rotation.
3. Measure q0 and q1; record two classical bits.
4. Send the bits to Bob over an ordinary channel.
5. Bob applies the X/Z correction the bits select.

Each of Alice's four outcomes occurs with probability 1/4 regardless of $|\psi ⟩$ — her measurement statistics leak nothing about the input. And the branch structure is the no-cloning theorem at work: the state shows up at Bob's side only because Alice's side gave it up. There is never a moment when both hold it.

Two myths to retire

First: teleportation is instant. It is not. Bob's reconstruction is gated on two classical bits traveling no faster than light, and before they arrive his qubit tells him nothing. The protocol transfers quantum state at the speed of a phone call plus a Bell pair.

Second: teleportation copies the input. It cannot — Chapter 21's theorem forbids it, and the protocol respects the theorem by construction: Alice's measurement destroys the original as the price of the transfer. A fax machine that burns the original is closer to the truth than a photocopier at a distance.

Teleportation as infrastructure

For a builder, teleportation is a distributed protocol with quantum and classical dependencies — not a gate. It earns its place as a primitive for quantum networking, modular architectures that move states between processors, and fault-tolerant circuit constructions, always with its costs attached.

The dependency graph every compiler, scheduler, or architecture memo must respect:

1. shared entanglement, prepared and distributed earlier;
2. a Bell measurement that consumes the input;
3. two classical bits in transit;
4. a conditional correction that finishes the reconstruction.

Delete any edge and the protocol fails: without fresh entanglement it degrades, without the classical bits it stalls, without the correction it outputs the wrong state confidently.

What a teleportation-based claim must show

When a networking or modular-computing pitch leans on teleportation, ask how entanglement is generated and at what rate, how its fidelity is maintained across distance or time, what the classical latency does to throughput, how corrections are applied, and which workload actually benefits.

Teleportation is real, demonstrated, and genuinely useful as a building block. It is also never free: the interesting engineering is always in the entanglement supply chain the pitch may prefer not to discuss.

Exercise

Trace, ledger, timeline. Draw the three-wire teleportation circuit and annotate where the unknown state, the Bell pair, the Bell measurement, the two classical bits, and the correction each appear. End with one sentence stating what infrastructure evidence a teleportation-based architecture claim must provide.

• Submit: the circuit sketch, a resource table, and a timeline marking the earliest moment Bob can reconstruct the state.
• Check: identify exactly where the measurement update changes Alice's registers, and explain why the original is not copied.
• Baseline: compare against sending a classical description of a known state, and explain why that route fails for an unknown arbitrary state.
• Repair: if your account implies instant transfer or object transport, rework Chapter 20 (Bell States and GHZ States) and Chapter 7 (Entanglement Without Faster-Than-Light Myths).

Check your understanding

Without notes: list the resources teleportation consumes, trace the corrections for all four of Alice's outcomes, and state when Bob's qubit first carries the input state.

Oral defense: someone calls teleportation "sending a qubit instantly." Correct each of the three errors in that sentence.

If you get stuck

If your protocol account omits the baseline, the classical bits, or the consumed entanglement, work back through Chapter 20 (Bell States and GHZ States) for the shared resource and Chapter 24 (Superdense Coding) for the ledger discipline — teleportation and superdense coding are the same machinery pointed in opposite directions.