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.

Measuring the transmitted qubit in the computational basis and forwarding its measured state produces a dephasing channel for users who do not know the outcome. For a Bell state $|{\Psi }^{+}⟩$, the reduced state becomes $(|01⟩⟨01|+|10⟩⟨10|)/2$. It retains perfect $Z$ anticorrelation while destroying $X$ correlation. Thus a single perfectly correlated key basis need not reveal an intercept-resend attack; incompatible-basis correlation tests can.

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.