Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 3 / 10D / Solution

Without Alice's operation, Bob obtains outcome

b

with probability

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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