Teleportation is usually presented with the shared pair in the state

PHYSICS · Multi-part · Fermi University Physics · Quantum Information & Technology · Quantum states, measurement and entanglement · Quantum states, measurement and entanglement

Quantum Information and Technology, Ch Entanglement as a resource

Problem

Teleportation is usually presented with the shared pair in the state ∣Φ+⟩|\Phi^+\rangle. Suppose instead Alice and Bob share ∣Ψ−⟩23=(∣01⟩23−∣10⟩23)/2|\Psi^-\rangle_{23}=(|01\rangle_{23}-|10\rangle_{23})/\sqrt2, and Alice holds an unknown qubit ∣ψ⟩1=a∣0⟩1+b∣1⟩1|\psi\rangle_1=a|0\rangle_1+b|1\rangle_1 that she wants to send to Bob. ((a)) Expand ∣ψ⟩1∣Ψ−⟩23|\psi\rangle_1|\Psi^-\rangle_{23} in the Bell basis of qubits 11 and 22, and show that Bob's qubit 33 collapses, for each of the four outcomes Φ+,Φ−,Ψ+,Ψ−\Phi^+,\Phi^-,\Psi^+,\Psi^- on qubits 1,21,2, to ∣ψ⟩|\psi\rangle up to one of the corrections I,X,Z,XZI,X,Z,XZ (up to an overall phase Bob need not correct). ((b)) Work the case ∣ψ⟩=∣+⟩|\psi\rangle=|+\rangle with measurement outcome Ψ+\Psi^+: state Bob's uncorrected qubit and show which correction from part (a) returns exactly ∣+⟩|+\rangle.

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