Teleportation: moving a quantum state

A transfer, with three clearly labelled qubits

Alice has an input qubit Q whose state must become available at Bob's module. Call its state |ψ⟩, pronounced 'psi'. Alice does not need to know a recipe for preparing it. Bob already holds a different physical qubit B. The task is to make B carry the input state, while preserving its quantum information. Neither a person nor a material object disappears and reappears.

For this lesson we assume ideal gates, measurements and storage. Alice and Bob first share the Bell pair (|00⟩ + |11⟩)/√2. Alice holds its qubit A; Bob holds its qubit B. Including the separate input Q, there are three qubits: Q and A at Alice, B at Bob. The Bell pair must have been established through a physical quantum process. An ordinary message alone cannot create it between the modules.

The operations, one step at a time

Alice applies a controlled-NOT gate with Q as the control and A as the target. On basis states, this flips A exactly when Q is 1. She next applies a Hadamard gate H to Q. These operations prepare her two qubits for the measurements required by the protocol. Measuring the untouched input directly and emailing the result would be a different procedure, and would lose information about an arbitrary input state.

Alice measures both of her qubits in the computational basis. We call the outcome from Q m1, and the outcome from A m2. She sends these two ordinary bits to Bob. The bit labels matter: swapping them would swap the correction rules. In the ideal protocol the four messages 00, 01, 10 and 11 are equally likely, regardless of the input state. These bits specify a correction; they are not a small classical description of |ψ⟩.

• Prepare the shared Bell pair; keep Q separate until the protocol begins.
• Apply CNOT from Q to A, then H to Q.
• Measure Q to obtain m1 and A to obtain m2; send both bits to Bob.
• Bob applies X if m2 = 1, then Z if m1 = 1.

Read the correction table

X exchanges |0⟩ and |1⟩. Z leaves |0⟩ unchanged and reverses the sign of the |1⟩ part; this changes a relative phase. Bob uses the message to undo the corresponding change to his state. Written compactly, the instruction is 'apply X^m2, then Z^m1'. An exponent 0 means skip that gate; an exponent 1 means apply it. In matrix notation the combined operation is Z^m1 X^m2 because the rightmost operation acts first.

Always read our messages in the order m1m2. For 00, Bob does nothing. For 01, he applies X. For 10, he applies Z. For 11, he applies X followed by Z. Each branch produces the same desired output state in the ideal model. The randomness selects the repair instruction; it does not select a different final answer after the repair.

What the protocol spends and what it preserves

One completed transfer consumes one shared Bell pair and communicates two classical bits in this standard protocol. Alice's original unknown state is no longer available on her measured qubit. The pair cannot simply be used again as the same entanglement resource. Another transfer requires another suitable resource, plus the necessary operations and communication.

Bob cannot use entanglement to read Alice's unknown state or a chosen message before her classical information arrives. Without that information his local measurement statistics reveal no usable signal from Alice. Thus teleportation does not communicate faster than light. The protocol can also transfer a qubit that is entangled with other qubits: ideal teleportation preserves those correlations, now involving Bob's output. That feature will let us move part of a computation in the next section.

Common misconceptions

A small glossary

Quantum teleportation

A protocol transferring quantum information using shared entanglement, local operations and classical communication.

Feed-forward

Using an earlier measurement result to choose a later operation.

Pauli X

The single-qubit gate exchanging the computational basis states |0⟩ and |1⟩.

Pauli Z

The single-qubit gate changing the sign of the |1⟩ amplitude relative to |0⟩.

Computational basis

The standard single-qubit measurement basis with possible outcomes labelled 0 and 1.