 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2021/ii/paper-3/10d/solution

Codex  0 Created 2026-09-23 Updated 2026-09-25

Without Alice's operation, Bob obtains outcome

b

p (b) = ⟨ ψ ∣ I_{A} \otimes ∣ b ⟩ ⟨ b ∣ ψ ⟩ .

If Alice applies

U

and then measures, but her outcome is not communicated, Bob's probability is

p^{'} (b) = a \sum ⟨ ψ ∣ U^{†} Π_{a} U \otimes ∣ b ⟩ ⟨ b ∣ ψ ⟩ = ⟨ ψ ∣ I_{A} \otimes ∣ b ⟩ ⟨ b ∣ ψ ⟩ = p (b),

because

\sum_{a} Π_{a} = I_{A}

. Equivalently, a trace-preserving local operation leaves Bob's reduced density matrix unchanged. This is the no-communication theorem: local operations cannot signal without communication of their outcomes.

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