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We consider quantum and private communications assisted by repeaters, fromthe basic scenario of a single repeater chain to the general case of anarbitrarily-complex quantum network, where systems may be routed through singleor multiple paths. In this context, we investigate the ultimate rates at whichtwo end-parties may transmit quantum information, distribute entanglement, orgenerate secret keys. These end-to-end capacities are defined by optimizingover the most general adaptive protocols that are allowed by quantum mechanics.Combining techniques from quantum information and classical network theory, wederive single-letter upper bounds for the end-to-end capacities in repeaterchains and quantum networks connected by arbitrary quantum channels,establishing exact formulas under basic decoherence models, including bosoniclossy channels, quantum-limited amplifiers, dephasing and erasure channels. Forthe converse part, we adopt a teleportation-inspired simulation of a quantumnetwork which leads to upper bounds in terms of the relative entropy ofentanglement. For the lower bounds we combine point-to-point quantum protocolswith classical network algorithms. Depending on the type of routing (single ormultiple), optimal strategies corresponds to finding the widest path or themaximum flow in the quantum network. Our theory can also be extended tosimultaneous quantum communication between multiple senders and receivers.

Capacities of repeater-assisted quantum communications