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ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2021/ii/paper-2/15d/b/solution

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Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 2 / 15D / b / Solution by

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

Alice adjoins

∣0 ⟩_{A^{'}}

and applies a local unitary whose action on the relevant inputs is

∣0 ⟩ ∣0 ⟩ \mapsto \frac{s}{t} ∣0 ⟩ ∣0 ⟩ + s q r t 1 - \frac{s ^{2}}{t ^{2}} ∣1 ⟩ ∣0 ⟩, ∣0 ⟩ ∣1 ⟩ \mapsto ∣0 ⟩ ∣1 ⟩ .

Measuring

A^{'}

and obtaining zero leaves the unnormalized state

s (∣00 ⟩ + ∣11 ⟩),

so the normalized state is the Bell state

∣ ϕ^{+} ⟩

. The success probability is

2 s^{2} > 0

.

Applying this entanglement concentration independently to

n

pairs yields an expected

2 s^{2} n

Bell pairs. Alice and Bob measure successful pairs in the computational basis to obtain identical secret random bits; public communication identifies successful positions without revealing outcomes. These bits form a secret key for a one-time pad of expected length

2 s^{2} n

.

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