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

Let Eve obtain $u,v\in \{\pm 1\}$. Applying part (b) to her axes gives $p(u,v)=[1-uv{\mathbf{n}}_a\cdot {\mathbf{n}}_b]/4$. Conditional on these rank-one local measurements, the forwarded particles are a product state with individual spin directions $u{\mathbf{n}}_a$ and $v{\mathbf{n}}_b$. HenceConditional independence gives the product of these means. Averaging over Eve's outcomes,The sum is $-{\mathbf{n}}_a\cdot {\mathbf{n}}_b$, soEve 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.