A Bell measurement is the projective measurement of two qubits in the orthonormal Bell state basis

∣ ϕ^{\pm} ⟩ = \frac{∣00 ⟩ \pm ∣11 ⟩}{2}, ∣ ψ^{\pm} ⟩ = \frac{∣01 ⟩ \pm ∣10 ⟩}{2} .

Its four classical outcomes identify the corresponding rank-one projector.

ii

Solution

Alice holds the unknown input

∣ α ⟩

and one qubit of a shared

∣ ϕ^{+} ⟩

pair; Bob holds the other. Alice performs a Bell measurement on her two qubits and sends its two-bit outcome to Bob. For outcomes

ϕ^{+}, ϕ^{-}, ψ^{+}, ψ^{-}

, Bob's qubit is respectively

∣ α ⟩, Z ∣ α ⟩, X ∣ α ⟩, XZ ∣ α ⟩

up to global phase. Bob applies respectively

I, Z, X, ZX

and obtains

∣ α ⟩

Replacing the resource by

(I \otimes U) ∣ ϕ^{+} ⟩

applies

U

to Bob's uncorrected output. If

P \in {I, Z, X, XZ}

labels the Bell outcome, the uncorrected state is

U P ∣ α ⟩

, and the unchanged correction produces

P^{- 1} U P ∣ α ⟩ .

Thus the four outputs are $U ∣ α ⟩$ , $Z U Z ∣ α ⟩$ , $X U X ∣ α ⟩$ , and $(XZ)^{- 1} U (XZ) ∣ α ⟩$ , up to global phases.

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