Past exam of the laser science and quantum information processing course of the University of Oxford / 2009 / 8 / 3 / Solution

Eve's computational-basis intercept-resend attack applies a projective measurement of $Z$ to Bob's photon and discards her outcome from Alice and Bob's description. Their resulting density matrix isThis separable state keeps $⟨Z\otimes Z⟩=-1$ but removes $⟨X\otimes X⟩$. HenceUsing the same signed Bell-test statistic givesThe ideal value is halved, and the attack no longer violates the CHSH inequality.

The key data show why several settings matter. For $(1,1)$, Eve's measurement leaves perfect anticorrelation, so this subset has zero bit errors after Bob flips his assignment. For $(3,2)$, $E=-1/2$, so correct anticorrelation has probability $(1-E)/2=3/4$: its quantum bit error rate is $1/4$. If both key-setting pairs occur equally often, their combined raw-key error rate is $1/8$. The Bell-test change detects this attack even though the vertical/horizontal key subset alone would not.