Entanglement swapping

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

Entanglement swapping, derived by teleporting qubit 22 through the resource pair, gives the identity ∣Φ+⟩12∣Φ+⟩34=12[∣Φ+⟩23∣Φ+⟩14+∣Φ−⟩23∣Φ−⟩14+∣Ψ+⟩23∣Ψ+⟩14+∣Ψ−⟩23∣Ψ−⟩14].\begin{aligned}&|\Phi^+\rangle_{12}|\Phi^+\rangle_{34}=\tfrac12\Big[|\Phi^+\rangle_{23}|\Phi^+\rangle_{14}+|\Phi^-\rangle_{23}|\Phi^-\rangle_{14}\\&+|\Psi^+\rangle_{23}|\Psi^+\rangle_{14}+|\Psi^-\rangle_{23}|\Psi^-\rangle_{14}\Big].\end{aligned} Suppose instead qubits 33 and 44 start in ∣Φ−⟩34=(I⊗Z)∣Φ+⟩34|\Phi^-\rangle_{34}=(I\otimes Z)|\Phi^+\rangle_{34} (with ZZ acting on qubit 44), while qubits 11 and 22 still share ∣Φ+⟩12|\Phi^+\rangle_{12}. You may use the facts that, with ZZ acting on the second qubit of each pair, Z∣Φ+⟩=∣Φ−⟩Z|\Phi^+\rangle=|\Phi^-\rangle, Z∣Φ−⟩=∣Φ+⟩Z|\Phi^-\rangle=|\Phi^+\rangle, Z∣Ψ+⟩=−∣Ψ−⟩Z|\Psi^+\rangle=-|\Psi^-\rangle and Z∣Ψ−⟩=−∣Ψ+⟩Z|\Psi^-\rangle=-|\Psi^+\rangle. ((a)) By applying I⊗ZI\otimes Z to qubit 44 on both sides of the identity above, derive the swapping identity for ∣Φ+⟩12∣Φ−⟩34|\Phi^+\rangle_{12}|\Phi^-\rangle_{34}. ((b)) State which single Bell state of qubits 11 and 44 results from each of the four possible Bell-basis outcomes on qubits 22 and 33, and confirm that qubits 11 and 44 end up in a definite (heralded) Bell pair every time, even though they have never interacted.

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