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ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2023/ii/paper-1/10d/b/solution

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Past exam of the mathematics course of the University of Cambridge / 2023 / ii / Paper 1 / 10D / b / Solution by

Codex  0 Created 2026-09-24 Updated 2026-09-25

Run the perfect discriminator on the supplied state and record whether the result is

i = 0

or

i = 1

. For each

i

, choose a unitary

V_{i}

satisfying

V_{i} ∣0 ⟩ = ∣ ϕ_{i} ⟩,

which is possible by extending

∣ ϕ_{i} ⟩

to an orthonormal basis. Prepare two fresh systems in

∣0 ⟩ ∣0 ⟩

and, controlled by the classical outcome

i

, apply

V_{i} \otimes V_{i}

. The output is

∣ ϕ_{i} ⟩ ∣ ϕ_{i} ⟩ .

The discriminator may destroy its input; the two freshly prepared outputs still implement cloning on this known pair. This is the construction that perfect discrimination implies cloning for a known state family.

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