Superdense coding is usually derived 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

Superdense coding is usually derived 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, Alice holding qubit 22 and Bob qubit 33. ((a)) Calculate (σ⊗I)∣Ψ−⟩23(\sigma\otimes I)|\Psi^-\rangle_{23}, up to global phase, for each σ∈{I,X,Z,XZ}\sigma\in\{I,X,Z,XZ\} applied to qubit 22, and identify each result as one of the four Bell states. ((b)) Explain why the four results remain mutually orthogonal, and state whether the resource inequality "11 qubit + 1+\,1 ebit ≥2\ge 2 bits" still holds when ∣Ψ−⟩|\Psi^-\rangle rather than ∣Φ+⟩|\Phi^+\rangle is the shared pair.

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