Intercept-resend attack

An eavesdropper measures quantum carriers and forwards states consistent with the outcomes. The resulting disturbance is tested statistically by the legitimate users. For a singlet measured by Eve on both sides along ${\mathbf{n}}_a,{\mathbf{n}}_b$, the forwarded correlation is $-(\mathbf{a}\cdot {\mathbf{n}}_a)({\mathbf{n}}_a\cdot {\mathbf{n}}_b)(\mathbf{b}\cdot {\mathbf{n}}_b)$. It follows by averaging the product of the conditional local outcome means over Eve's correlated outcomes. This model does not cover every general quantum attack.

Use Alice's CHSH axes $\hat{x},\hat{y}$ and Bob's $(\hat{x}+\hat{y})/\sqrt{2},(-\hat{x}+\hat{y})/\sqrt{2}$, with sign pattern $S=C_{11}-C_{13}+C_{31}+C_{33}$. If Eve measures both singlet particles along planar angles $\alpha ,\beta$, the intercept-resend attack correlation gives $S=-\sqrt{2}{\cos }^2(\alpha -\beta )$. Consequently $-\sqrt{2}\le S\le 0$. Randomly mixing axes independently of the users' secret settings preserves this interval by averaging. The bound is specific to these settings and this attack, rather than a general security theorem.