Quantum and private communications are affected by a fundamental limitation which severely restricts the optimal rates that are achievable by two distant parties. To overcome this problem, one needs to introduce quantum repeaters and, more generally, quantum communication networks. Within a quantum network, other problems and features may appear when we move from the basic unicast setting of single-sender/single-receiver to more complex multiend scenarios, where multiple senders and multiple receivers simultaneously use the network to communicate. Assuming various configurations, including multiple-unicast, multicast, and multiple-multicast communication, we bound the optimal rates for transmitting quantum information, distributing entanglement, or generating secret keys in quantum networks connected by arbitrary quantum channels. These bounds cannot be surpassed by the most general adaptive protocols of quantum network communication.
A. Adaptive point-to-point protocolsLet us first discuss the general structure of an adaptive point-to-point protocol P through a quantum channel E, following the notation from Ref. . Alice has a local register of quantum systems a and Bob has another local register b; these are prepared in a state ρ 0 ab by means