In Quantum Key distribution, it is my understanding that when Eve tries to intercept photons being sent from Alice to Bob, she has a 50% chance of using the correct polarization filter, and so 50% chance of getting the correct bit information. Let's just say hypothetically that she somehow gets all the photons she intercepted using the correct filter (and hence correct bit information), and re emits them so that they're all exactly the same photons as Alice had intended them to be.
Doesn't this mean that during the error reconciliation between Alice and Bob, they'll detect no errors and hence conclude that the distribution of the key was secure, but in reality, Eve has perfect information about the "secret" key? Doesn't this violate the proof that QKD is "provably secure"?