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).