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arXiv:2511.10584 · PRX Quantum

Finite-size quantum key distribution rates from Rényi entropies using conic optimization

Mariana Navarro, Andrés González Lorente, Pablo V. Parellada, Carlos Pascual-García, Mateus Araújo

quant-phSubmitted Nov 13, 2025• Announced Nov 14, 2025• 0 citations• Revised Jan 30, 2026 (v2)
v1 · Nov 13, 2025v2 · Jan 30, 2026v3 · Apr 16, 2026v4 · Oct 14, 2026
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Abstract

Finite-size general security proofs for quantum key distribution based on Rényi entropies have recently been introduced. These approaches are more flexible and provide tighter bounds on the secret key rate than traditional formulations based on the von Neumann entropy. However, deploying them requires minimizing the conditional Rényi entropy, a difficult optimization problem that has hitherto been tackled using ad-hoc techniques based on the Frank-Wolfe algorithm, which are unstable and can only handle particular cases. In this work, we introduce a method based on non-symmetric conic optimization for solving this problem. Our technique is fast, reliable, and completely general. We illustrate its performance on several protocols, whose results represent an improvement over the state of the art.

Journal reference
PRX Quantum 7, 033051 (2026)
DOI
10.1103/bf9s-m4jb
Comments
40 pages, 5 figures. Added section 3.4 and extended Remark 1

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Finite-size general security proofs for quantum key distribution based on Rényi entropies have recently been introduced. These approaches are more flexible and provide tighter bounds on the secret key rate than traditional formulations based on the von Neumann entropy. However, deploying them requires minimizing the conditional Rényi entropy, a difficult optimization problem that has hitherto been tackled using ad-hoc techniques based on the Frank-Wolfe algorithm, which are unstable and can only handle particular cases. In this work, we introduce a method based on non-symmetric conic optimization for solving this problem. Our technique is fast, reliable, and completely general. We illustrate its performance on several protocols, whose results represent an improvement over the state of the art.
Quantum key distribution (QKD) is a form of secure communication which transmits secret keys between two trusted parties. Evaluation of the achievable secret key rates are essential to ensure robust security of any QKD protocols. As such, the optimization of the secret key rate to obtain tight and reliable bounds are required especially for instances when the number of transmitted signals are finite. In this work, we make use of the generalized Rényi entropic quantities to obtain a tighter bound on the key rate compared to previous works which uses the von Neumann entropy. However, the optimization of Rényi entropic quantities is often non-trivial. Hence, we derive a generalized bound on the Rényi entropy in terms of the Rényi divergence which allows us to apply the Frank-Wolfe minimization algorithm for the optimization of secret key rates, thus generalizing existing numerical frameworks. With this generalized framework, we show improvements in regimes where finite signals experience high loss, thus ensuring relevance for real-world scenarios, particularly for satellite-based QKD implementations.
Computing key rates in quantum key distribution (QKD) numerically is essential to unlock more powerful protocols, that use more sophisticated measurement bases or quantum systems of higher dimension. It is a difficult optimization problem, that depends on minimizing a convex non-linear function: the (quantum) relative entropy. Standard conic optimization techniques have for a long time been unable to handle the relative entropy cone, as it is a non-symmetric cone, and the standard algorithms can only handle symmetric ones. Recently, however, a practical algorithm has been discovered for optimizing over non-symmetric cones, including the relative entropy. Here we adapt this algorithm to the problem of computation of key rates, obtaining an efficient technique for lower bounding them. In comparison to previous techniques it has the advantages of flexibility, ease of use, and above all performance.
Quantum key distribution requires tight and reliable bounds on the secret key rate to ensure robust security. This is particularly so for the regime of finite block sizes, where the optimization of generalized Rényi entropic quantities is known to provide tighter bounds on the key rate. However, such an optimization is often non-trivial, and the non-monotonicity of the key rate in terms of the Rényi parameter demands additional optimization to determine the optimal Rényi parameter as a function of block sizes. In this work, we present a tight analytical bound on the Rényi entropy in terms of the Rényi divergence and derive the analytical gradient of the Rényi divergence. This enables us to generalize existing state-of-the-art numerical frameworks for the optimization of the key rate. With this generalized framework, we show improvements in regimes of high loss and low block sizes, which are particularly relevant for long-distance satellite-based protocols.
A realistic quantum-key-distribution protocol necessarily runs with finite resources. Usually, security proofs for existing quantum key distribution are asymptotic in the sense that certain parameters are exceedingly large compared to practical realistic values. In this paper, we derive bounds on the secret key rates for the Bennett 1992 protocol, which includes a preprocessing step. The derivation for a finite-size key is expressed as an optimization problem by using results from the uncertainty relations and the smooth R\'enyi entropies.
Computing the key rate in quantum key distribution (QKD) protocols is a long standing challenge. Analytical methods are limited to a handful of protocols with highly symmetric measurement bases. Numerical methods can handle arbitrary measurement bases, but either use the min-entropy, which gives a loose lower bound to the von Neumann entropy, or rely on cumbersome dedicated algorithms. Based on a recently discovered semidefinite programming (SDP) hierarchy converging to the conditional von Neumann entropy, used for computing the asymptotic key rates in the device independent case, we introduce an SDP hierarchy that converges to the asymptotic secret key rate in the case of characterised devices. The resulting algorithm is efficient, easy to implement and easy to use. We illustrate its performance by recovering known bounds on the key rate and extending high-dimensional QKD protocols to previously intractable cases. We also use it to reanalyse experimental data to demonstrate how higher key rates can be achieved when the full statistics are taken into account.