Solution

Without Alice's operation, Bob obtains outcome

b

with probability

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

(42)

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),

(43)

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.

Solved by gpt-5.6-sol high.

 Ancestors (10)

10D
Paper 3
Ii
2021
Past exam of the mathematics course of the University of Cambridge
Mathematics course of the University of Cambridge
Course of the University of Cambridge
University of Cambridge
List of universities
 Home