Security of Quantum Key Distribution with Entangled Photons Against   Individual Attacks
  Security of the Ekert protocol is proven against individual attacks where an eavesdropper is allowed to share any density matrix with the two communicating parties. The density matrix spans all of the photon number states of both receivers, as well as a probe state of arbitrary dimensionality belonging to the eavesdropper. Using this general eavesdropping strategy, we show that the Shannon information on the final key, after error correction and privacy amplification, can be made exponentially small. This is done by finding a bound on the eavesdropper's average collision probability. We find that the average collision probability for the Ekert protocol is the same as that of the BB84 protocol for single photons, indicating that there is no analog in the Ekert protocol to photon splitting attacks. We then compare the communication rate of both protocols as a function of distance, and show that the Ekert protocol has potential for much longer communication distances, up to 170km, in the presence of realistic detector dark counts and channel loss. Finally, we propose a slightly more complicated scheme based on entanglement swapping that can lead to even longer distances of communication. The limiting factor in this new scheme is the fiber loss, which imposes very slow communication rates at longer distances. 
Edo Waks, Assaf Zeevi, and Yoshihisa Yamamoto