 Solution

ID: past-exam-of-the-computer-science-course-of-the-university-of-oxford/2020/quantum-information/1/d/solution

The map sends

∣0 ⟩

∣0 ⟩

∣1 ⟩

∣2 ⟩

, and

∣2 ⟩

λ ∣1 ⟩

. These output columns are mutually orthogonal, with squared norms

1, 1, ∣ λ ∣^{2}

. Consequently

M_{λ}^{†} M_{λ} = diag (1, 1, ∣ λ ∣^{2}),

and

M_{λ} is unitary if and only if ∣ λ ∣ = 1, equivalently λ = e^{i θ} for real θ .

Necessity follows from preserving the norm of

∣2 ⟩

; sufficiency follows because the three columns are then an orthonormal basis. Merely requiring

λ \neq = 0

would establish invertibility but not unitarity. This is the weighted permutation unitarity criterion.

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