Quantum Teleportation

Suppose Alice wants to send an arbitrary quantum state |ψ⟩ = α|0⟩ + β|1⟩ to Bob. Directly transmitting the qubit might be impossible or unreliable if the channel is noisy. Instead, Alice and Bob share one qubit each of a maximally entangled Bell pair |Φ⁺⟩ = (|00⟩ + |11⟩)/√2. Alice holds both the unknown state |ψ⟩ and her half of the Bell pair; Bob holds the other half.

Alice performs a Bell state measurement on her two qubits: she applies a CNOT gate from the unknown qubit to her entangled qubit, followed by a Hadamard gate on the unknown qubit. She then measures both qubits in the computational basis, obtaining two classical bits b1 and b2. These two bits completely characterize the relationship between |ψ⟩ and Bob's qubit.

Alice sends the two classical bits to Bob over any classical channel. Based on these bits, Bob applies corrective operations to his qubit: if b2 = 1 he applies X, and if b1 = 1 he applies Z. After these corrections, Bob's qubit is exactly |ψ⟩, regardless of which measurement outcome Alice obtained. The original state on Alice's side is destroyed by the measurement, consistent with the no-cloning theorem.

• •Shared Entanglement: Alice and Bob must each hold one qubit of a Bell pair before the protocol begins.
• •Bell Measurement: Alice entangles |ψ⟩ with her half of the Bell pair and measures both qubits.
• •Classical Communication: Two classical bits must be transmitted from Alice to Bob, preventing faster-than-light communication.
• •Corrections: Bob applies Pauli X and/or Z gates based on the classical bits to recover |ψ⟩ exactly.

The full three-qubit state can be rewritten in the Bell basis of Alice's two qubits. After Alice applies CNOT and Hadamard, the state becomes a superposition over the four Bell states, each correlated with a specific Pauli-transformed version of |ψ⟩ on Bob's qubit. When Alice measures, she collapses the state to one of four possibilities, and Bob's qubit instantaneously becomes X^b2 Z^b1 |ψ⟩.

The four possible measurement outcomes (00, 01, 10, 11) each occur with probability 1/4, regardless of α and β. This means Alice learns nothing about the state |ψ⟩ from her measurement — the information is perfectly encrypted until Bob applies the correction. The protocol is therefore both universal (works for any state) and information-theoretically secure against eavesdroppers who only intercept the classical bits.

• •Bell Basis Decomposition: The total state is an equal superposition of four terms, each pairing a Bell state with a corrected version of |ψ⟩.
• •Measurement Outcomes: Each of the four outcomes occurs with probability 1/4, independent of the amplitudes α and β.
• •Correction Table: 00 → I, 01 → X, 10 → Z, 11 → ZX. Bob applies the appropriate gate based on the bits received.