Alice and Bob share the product state and a telephone line

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

Alice and Bob share the product state ∣0⟩A∣0⟩B|0\rangle_A|0\rangle_B and a telephone line, with no fibre or other link carrying qubits between them. Alice applies a Hadamard gate to her qubit and measures it in the computational basis, obtaining outcome m∈{0,1}m\in\{0,1\} with probability 12\tfrac12 each. She telephones mm to Bob, who applies XmX^m to his own qubit. Someone argues: 'Averaged over Alice's random outcome, the two qubits end up perfectly correlated, both 00 or both 11 with probability 12\tfrac12 each. That is exactly the correlation the Bell state ∣Φ+⟩|\Phi^+\rangle gives when both qubits are measured in the computational basis. So this LOCC procedure has created an entangled pair.' ((a)) Write down the actual state of the pair after the protocol, as a density operator, without assuming the conclusion. ((b)) Compute ⟨Z⊗Z⟩\langle Z\otimes Z\rangle and ⟨X⊗X⟩\langle X\otimes X\rangle for this state and for ∣Φ+⟩|\Phi^+\rangle. ((c)) Decide whether the state is entangled, and use part (b) to identify the error in the argument.

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