Photon loss in optical channels fundamentally limits long-range reliable quantum communication. A standard approach to overcoming this limitation is the use of quantum repeater nodes, which typically perform experimentally demanding non-Gaussian operations. However, whether Gaussian repeater protocols can enhance quantum communication rates over bosonic attenuation channels has remained open. In this Letter, we prove a no-go theorem for Gaussian quantum repeaters in a quantum network. Specifically, we show that a repeater chain composed of Gaussian operations, homodyne measurements, and arbitrary classical communication cannot enhance the quantum and private capacity of a pure-loss attenuation channel beyond that achievable by direct transmission. In fact, arbitrary Gaussian local operation and classical communication among the repeater nodes cannot enhance the two-way capacities either. Our proof introduces a generalization of $k$ extendibility to a notion of fractional extendibility for Gaussian states and establishes some of its useful properties, thereby providing a powerful framework for analyzing Gaussian quantum networks.

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• Quantum communication
• Quantum information theory
• Quantum networks
• Quantum repeaters