= Solution
Let Eve obtain $u,v\in\{\pm1\}$. 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 $v\mathbf n_b$. Hence
$$
\mathbb E[X\mid u]=u\mathbf a\cdot\mathbf n_a,\qquad
\mathbb E[Y\mid v]=v\mathbf b\cdot\mathbf n_b.
$$
Conditional independence gives the product of these means. Averaging over Eve's outcomes,
$$
C=(\mathbf a\cdot\mathbf n_a)(\mathbf b\cdot\mathbf n_b)
\sum_{u,v}uv\frac{1-uv\mathbf n_a\cdot\mathbf n_b}{4}.
$$
The sum is $-\mathbf n_a\cdot\mathbf n_b$, so
$$
\boxed{C(\mathbf a,\mathbf b)=-(\mathbf a\cdot\mathbf n_a)
(\mathbf n_a\cdot\mathbf n_b)(\mathbf b\cdot\mathbf n_b).}
$$
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.
Back to article page
