First, we look at the case where she measures both photons. It is clear that she has an opportunity here to learn much more about the bit sent than in the single photon case. In fact, if she measures one photon in one of the two bases used for BB84 encoding and the other photon in the other basis, she will learn the bit with certainty after the basis has been revealed. Doing so, Eve will introduce noise on Bob's side. We want to find the minimum probability for Eve to introduce an error after measuring the two photons and having prepared a strong pulse according the outcome of her measurement.We call this a Double Intercept-Resend attack (DIR).
The maximum probability of guessing correctly the bit (for all possible measurements not only individual ones) acting on two identically polarized BB84 qubits can easily be obtained from the density matrices and for the encoding of bits 0 and 1 respectively using the formula . Carrying out the calculations give . The maximum probability of guessing correctly the tranmission basis can be computed similarly giving . Given those conditions on and , we now find the best possible measurement for determining the state of the strong pulse that Eve resends to Bob.
Let be the probability of guessing the bit correctly
given the guess
on the basis is correct and let
be the probability of guessing the bit
given the guess for the basis is wrong. We have that