The total transmission of the line is the product of two contributions:
The photon statistics for the pulses generated by our laser is close to the Poissonian distribution described by the parameter which is an average photon number in a pulse sent by Alice.
The probability to detect a photon at Bob's side, with no dark counts taken into account, is then
Since it is impossible to distinguish between dark counts and regular counts, they both contribute to the probability of a detection event . The coincidence term is ignored here. is the implementation efficiency, which is equal to the exploitation of the pulses, that is for BB84, for BB84 implemented with only one detector1 and for B92.
The contributions to the errors are the same as the contributions to
the counts, i.e. ignoring the coincidence term, we get for the error
and is the probability of an error due
to the signal and dark counts respectively. The error rate
is given by the probability of an error divided by the probability of
a detection event:
In the following section we consider, one, two and many photon attacks separately. Finally the results are combined to obtain optimal conditions on the pulse strength , quantum efficiency and the dark count probability necessary to maximize the secure bit rate.