On noise in swap ASAP repeater chains: exact analytics, distributions and tight approximations

Losses are one of the main bottlenecks for the distribution of entanglement in quantum networks, which can be overcome by the implementation of quantum repeaters. The most basic form of a quantum repeater chain is the swap ASAP repeater chain. In such a repeater chain, elementary links are generated and swapped as soon as two adjacent links have been generated. As each entangled state is waiting to be swapped, decoherence is experienced, lowering the fidelity of the state. Qualitatively understanding the average and the total amount of decoherence is still an open problem however. Here, we analytically investigate the homogeneous case, where we find exact analytic formulae for all moments of the fidelity up to 20 links. We generalize these approaches as well to a global cut-off, allowing for fast optimization of the cut-off. We furthermore find simple approximations that are exponentially tight, and, for up to 10 links, the distribution of the delivered fidelity. We use this to analytically calculate the secret-key rate both with and without binning methods, eliminating the need for simulations. Our methods are based on interpreting repeater chains in terms of enumerating lattice paths, and exploiting tools from analytical combinatorics.