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ID: past-exam-of-the-quantum-information-course-of-the-university-of-cambridge/2015/paper-1/2/d/solution

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Past exam of the quantum information course of the University of Cambridge / 2015 / Paper 1 / 2 / d / Solution by

Codex  0 2026-10-20

Let Eve obtain

u, v \in {\pm 1}

. Applying part (b) to her axes gives

p (u, v) = [1 - uv n_{a} \cdot n_{b}] /4

. Conditional on these rank-one local measurements, the forwarded particles are a product state with individual spin directions

u n_{a}

and

v n_{b}

. Hence

E [X ∣ u] = u a \cdot n_{a}, E [Y ∣ v] = v b \cdot n_{b} .

Conditional independence gives the product of these means. Averaging over Eve's outcomes,

C = (a \cdot n_{a}) (b \cdot n_{b}) \sum_{u, v} uv \frac{1 - uv n _{a} \cdot n _{b}}{4} .

The sum is

- n_{a} \cdot n_{b}

, so

C (a, b) = - (a \cdot n_{a}) (n_{a} \cdot n_{b}) (b \cdot n_{b}) .

Eve has turned the entangled source into an ensemble of product states by an intercept-resend attack. Ignoring her records does not restore the original singlet coherence.

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