Relaying Quantum Information

Quantum relays are central to both quantum communication and distributed quantum computing, enabling long-distance transmission and modular architectures. Unlike classical repeaters, quantum repeaters preserve coherence without amplifying quantum information, relying on entanglement swapping and quantum error correction to overcome loss and decoherence. In this work, we investigate the transmission of quantum information via quantum relay channels. Our three-terminal relay model captures the trade-off between repeater-assisted and repeaterless communication strategies. Specifically, we propose a partial decode-forward strategy, in which quantum ``message system" consists of two components. The first component is decoded by the relay and then sent to the destination receiver, whereas the second component is decoded by the destination receiver without the relay's help. We analyze both entanglement-assisted and unassisted scenarios. As a special case, the full decode-forward strategy is recovered, with the relay decoding, re-encoding, and forwarding the entire message. Our framework allows for different entanglement topologies between the transmitter, the relay and the destination receiver, recovering known results on entanglement-assisted and unassisted communication. Furthermore, we discuss the interpretation of coding with quantum side information. These findings provide a foundation for designing secure, efficient, and reliable quantum networks and for realizing practical quantum repeaters and long-range quantum key distribution.