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Maximum tolerable excess noise in continuous-variable quantum key distribution and improved lower bound on two-way capacities

Nature Photonics volume 19, pages 329–334 (2025)Cite this article

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Abstract

The two-way capacities of quantum channels determine the ultimate entanglement and secret-key distribution rates achievable by two distant parties that are connected by a noisy transmission line, in the absence of quantum repeaters. Since repeaters will likely be expensive to build and maintain, a central open problem of quantum communication is to understand what performances are achievable without them. Here we find a new lower bound on the energy-constrained and unconstrained two-way quantum and secret-key capacities of all phase-insensitive bosonic Gaussian channels, namely thermal attenuator, thermal amplifier and additive Gaussian noise, which are realistic models for the noise affecting optical fibres or free-space links. Ours is the first non-zero lower bound on the two-way quantum capacity in the parameter range where the (reverse) coherent information becomes negative, and it shows explicitly that entanglement distribution is always possible when the channel is not entanglement breaking. This completely solves a crucial open problem of the field, namely establishing the maximum excess noise, which is tolerable in continuous-variable quantum key distribution. In addition, our construction is fully explicit; that is, we devise and optimize a concrete entanglement distribution and distillation protocol that works by combining recurrence and hashing protocols.

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Fig. 1: Bounds on the two-way capacities of the thermal attenuator, thermal amplifier and additive Gaussian noise.

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Acknowledgements

F.A.M. and V.G. acknowledge financial support by MUR (Ministero dell’Istruzione, dell’Università e della Ricerca) through the following projects: PNRR MUR project PE0000023-NQSTI, PRIN 2017 Taming complexity via Quantum Strategies: a Hybrid Integrated Photonic approach (QUSHIP) Id. 2017SRN-BRK, and project PRO3 Quantum Pathfinder. L.L. was partially supported by the Alexander von Humboldt Foundation. F.A.M. and L.L. thank the Freie Universität Berlin for hospitality. F.A.M., L.L. and V.G. thank D. Miller, M. Rosati, M. Notarnicola, M. Avesani, M. Mazelanik and M. Barbieri for useful discussions.

Author information

Authors and Affiliations

  1. NEST, Scuola Normale Superiore and Istituto Nanoscienze, Pisa, Italy

    Francesco Anna Mele

  2. QuSoft, Amsterdam, the Netherlands

    Ludovico Lami

  3. Korteweg-de Vries Institute for Mathematics, University of Amsterdam, Amsterdam, the Netherlands

    Ludovico Lami

  4. Institute for Theoretical Physics, University of Amsterdam, Amsterdam, the Netherlands

    Ludovico Lami

  5. Institut für Theoretische Physik und IQST, Universität Ulm, Ulm, Germany

    Ludovico Lami

  6. NEST, Scuola Normale Superiore and Istituto Nanoscienze, Consiglio Nazionale delle Ricerche, Pisa, Italy

    Vittorio Giovannetti

Authors
  1. Francesco Anna Mele
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  2. Ludovico Lami
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  3. Vittorio Giovannetti
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Contributions

The entanglement distribution protocol was designed and optimized by F.A.M. The proof of Theorem 1 was found in a blackboard discussion between the three authors. F.A.M. wrote a first complete draft of the paper, which was subsequently improved by L.L. and V.G.

Corresponding authors

Correspondence to Francesco Anna Mele, Ludovico Lami or Vittorio Giovannetti.

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Nature Photonics thanks Javier Fonollosa and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

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Supplementary Information

36 pages of detailed derivations of our results.

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Mele, F.A., Lami, L. & Giovannetti, V. Maximum tolerable excess noise in continuous-variable quantum key distribution and improved lower bound on two-way capacities. Nat. Photon. 19, 329–334 (2025). https://doi.org/10.1038/s41566-024-01595-9

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