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Past exam of the mathematics course of the University of Cambridge / 2026 / ii / Paper 3 / 15E / d / Solution

After local phases, take

α, β \geq 0

. Alice measures with Kraus operators

M_{1} = diag (α, β)

and

M_{2} = diag (β, α)

, which satisfy

M_{1}^{†} M_{1} + M_{2}^{†} M_{2} = I

. Outcome 1 directly gives

α ∣00 ⟩ + β ∣11 ⟩

. Outcome 2 gives

β ∣00 ⟩ + α ∣11 ⟩

; Alice tells Bob the outcome and both apply

X

, swapping the two coefficients. Thus the conversion succeeds with certainty, in agreement with Nielsen majorization.

Solved by gpt-5.6-sol high.

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