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 [8]. 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

for some choice of . Eve cannot do better than optimizing (5) subject to (4) and even if she was allowed to implement any measurement on two qubits. One can verify that the maximum of (5) occurs whenever and leading to

The lower bound on holds even if Eve could perform any quantum measurement providing the state of the strong signal she then forwards to Bob. The bound is conservative since it would be surprising that a single quantum measurement could provide guesses for both the bit and the basis each reaching the maximum.