**Mathematical model for practical entanglement swapping and application to long-distance quantum key distribution**

Entanglement swapping (ES) between photon pairs is a key building block
in entanglement-based quantum communication schemes using quantum
relays or quantum repeaters to overcome the range limits of
long-distance quantum key distribution (QKD). Its practical
realization, however, suffers from real-world imperfections.
In our recent work [Scherer et al., Phys. Rev. A 80, 062310 (2009)],
we have developed a nonperturbative mathematical model for practical
ES which accounts for detector inefficiencies, detector dark counts
and the unavoidable multipair events of common realistic sources
of entangled photon pairs. Our closed-form solution allows us to
calculate the "amount" of entanglement after a realistic ES
depending on parameters that are directly related to an experiment,
such as dark-count rates and detection efficiencies of off-the-shelf
detectors as well as brightness of parametric down-conversion sources.
We now demonstrate that our model is useful for planning long-distance
QKD experiments.
Here we expand on the impact of real-world imperfections on QKD
performance in schemes aiming at distribution of shared photon entanglement
over longer distances by means of an ES operation. In particular, our
analysis provides the optimal trade-off between detector efficiency and
dark counts (which are usually not independent) as well as the optimal
photon-pair production rate (brightness) of the sources that maximize
the secret key rate for a given distance between a sender (Alice)
and a receiver (Bob).