# Eavesdropping in case of the BB84 Protocol

I try to understand eavesdropping in case of the BB84 protocol (lets assume we have single photons, no noise etc.). Alice and Bob generate a classical random bit $$a_i'$$ and $$b_i'$$. Alice generates an encoded quantum bit $$a_i$$ based on $$a_i'$$ and sends it to Bob. Bob decodes $$a_i$$ based on the $$b_i'$$. Alice and Bob share $$a_i'$$ and $$b_i'$$ on a classical channel. If they used the same encoding/decoding (i.e. $$a_i' = b_i'$$) the decoded bit can be used, otherwise it will be ignored. Now, Eve tries eavesdropping. She would measure $$a_i$$ based on a random base. Her guess will be correct by a chance of 50 %. In case she used the wrong base she will still measure the correct value by 50 % change. In total, by 25 % chance she would not measure the correct value. Alice and Bob can monitor thier error rate and if it is too high there is a high probability of eavesdropping.

But I do have to questions regaring eavesdropping:

1. What if Eve generates an entangled Qubit of $$a_i$$? Eve could wait to measure $$a_i$$ until Alice and Bob exchange $$a_i'$$ and $$b_i'$$ (in this case it would not be a problem that the superposition of $$a_i$$ already collapsed because Eve would know the correct base). Since Eve could know how to decode/encode based on $$a_i'$$ and $$b_i'$$ she could reconstruct $$a_i$$.

2. Moreover, what if Eve behaves like a fake Alice and Bob? That is, Eve pretends to be Bob when communicating with Alice. Once she decoded the information she could send it to Bob while she would pretend to be Alice.