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A potential quantum internet would open up the possibility of realizing numerous new applications, including provably secure communication. Since losses of photons limit long-distance, direct quantum communication and wide-spread quantum networks, quantum repeaters are needed. The so-called PLOB-repeaterless bound [Pirandola et al., Nat. Commun. 8, 15043 (2017)] is a fundamental limit on the quantum capacity of direct quantum communication. Here, we analytically derive the quantum-repeater gaindoi:10.22331/q-2019-12-16-216 fatcat:dp6fnyc2ofe7hpi25ke5pa2ugq