Practical long-distance quantum communication using concatenated entanglement swapping

We construct a theory for long-distance quantum communication based on sharing entanglement through a linear chain of N elementary swapping segments of length L=Nl where l is the length of each elementary swap setup. Entanglement swapping is achieved by linear optics, photon counting, and postselection, and we include effects due to multiphoton sources, transmission loss, and detector inefficiencies and dark counts. Specifically we calculate the resultant four-mode state shared by the two parties at the two ends of the chain, and we derive the two-photon coincidence rate expected for this state and thereby the visibility of this long-range-entangled state. The expression is a nested sum with each sum extending from zero to infinite photons, and we solve the case N=2 exactly for the ideal case (zero dark counts, unit-efficiency detectors, and no transmission loss) and numerically for N=2 in the nonideal case with truncation at n₌₀ₗ=3 photons in each mode. For the general case, we show that the computational complexity for the numerical solution is n₌₀ₗ^12N.