  We analyze an entanglement-based quantum key distribution (QKD) architecture
that uses a linear chain of quantum repeaters employing photon-pair sources,
spectral-multiplexing, linear-optic Bell-state measurements, multi-mode quantum
memories and classical-only error correction. Assuming perfect sources, we find
an exact expression for the secret-key rate, and an analytical description of
how errors propagate through the repeater chain, as a function of various loss
and noise parameters of the devices. We show via an explicit analytical
calculation, which separately addresses the effects of the principle
non-idealities, that this scheme achieves a secret key rate that surpasses the
TGW bound---a recently-found fundamental limit to the rate-vs.-loss scaling
achievable by any QKD protocol over a direct optical link---thereby providing
one of the first rigorous proofs of the efficacy of a repeater protocol. We
explicitly calculate the end-to-end shared noisy quantum state generated by the
repeater chain, which could be useful for analyzing the performance of other
non-QKD quantum protocols that require establishing long-distance entanglement.
We evaluate that shared state's fidelity and the achievable entanglement
distillation rate, as a function of the number of repeater nodes, total range,
and various loss and noise parameters of the system. We extend our theoretical
analysis to encompass sources with non-zero two-pair-emission probability,
using an efficient exact numerical evaluation of the quantum state propagation
and measurements. We expect our results to spur formal rate-loss analysis of
other repeater protocols, and also to provide useful abstractions to seed
analyses of quantum networks of complex topologies.
