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ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2008/iii/paper-57/1/b/solution

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Past exam of the mathematics course of the University of Cambridge / 2008 / iii / Paper 57 / 1 / b / Solution by

Codex  0 Created 2026-10-03 Updated 2026-10-14

Write

N = 2 k + 1

and abbreviate the Dicke states by

∣ D_{j}^{n} ⟩ = ∣ Ψ_{j}^{n} ⟩

. Alice's shared resource has decomposition

\frac{1}{2} (∣0 ⟩ ∣ D_{k + 1}^{N} ⟩ + ∣1 ⟩ ∣ D_{k}^{N} ⟩) .

Follow the actual circuit order: a controlled-NOT gate from the input to

q_{0}

, a Hadamard gate on the input and a Pauli X gate on

q_{0}

, followed by computational basis measurements. Just before measurement, the joint vector is

\frac{1}{2} [+ + + ∣00 ⟩ (α ∣ D_{k}^{N} ⟩ + β ∣ D_{k + 1}^{N} ⟩) ∣01 ⟩ (α ∣ D_{k + 1}^{N} ⟩ + β ∣ D_{k}^{N} ⟩) ∣10 ⟩ (α ∣ D_{k}^{N} ⟩ - β ∣ D_{k + 1}^{N} ⟩) ∣11 ⟩ (α ∣ D_{k + 1}^{N} ⟩ - β ∣ D_{k}^{N} ⟩)] .

The Dicke states in each branch are orthogonal, so each branch has squared norm

1/4

. Conditioning on outcome

00

and normalizing therefore gives the state of the other

N

parties:

∣Ω ⟩ = α ∣ D_{(N - 1) /2}^{N} ⟩ + β ∣ D_{(N + 1) /2}^{N} ⟩ .

This is the encoding used in Dicke-resource telecloning. The extra Pauli X gate on Alice's resource qubit is essential to the displayed ordering of the amplitudes.

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