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Finite-size secret-key rates of discrete modulation continuous-variable quantum key distribution under Gaussian attacks

Gabriele Staffieri1,2, Giovanni Scala3,2, and Cosmo Lupo3,1,2

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Phys. Rev. A 113, 022445 – Published 24 February, 2026

DOI: https://doi.org/10.1103/rwq3-p1m6

Abstract

Quantum conditional entropies play a fundamental role in quantum information theory. In quantum key distribution, they are exploited to obtain reliable lower bounds on the secret-key rates in the finite-size regime, against collective attacks and coherent attacks under suitable assumptions. Here we consider continuous-variable communication protocols, where the sender Alice encodes information using a discrete modulation of phase-shifted coherent states, and the receiver Bob decodes by homodyne or heterodyne detection. We compute the Petz-Rényi and sandwiched Rényi conditional entropies associated with these setups, assuming either a passive eavesdropper or one that injects thermal photons into the channel, who gathers the quantum information leaked through a lossy communication line of known or bounded transmittance. Whereas our results do not directly provide reliable key-rate estimates, they do represent useful ballpark figures. We obtain analytical or semianalytical expressions that do not require intensive numerical calculations. These expressions serve as bounds on the key rates that may be tight in certain scenarios. We compare different estimates, including known bounds that have already appeared in the literature and bounds that we determine. The latter are found to be tighter for very short block sizes.

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Outline

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Phys. Rev. A 113, 022445– Published 24 February, 2026
  • Received 17 September 2025
  • Revised 5 January 2026
  • Accepted 2 February 2026
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

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