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arXiv:2608.23285

Efficient Computation of QKD Key Rates without Semidefinite Programming

Bence Temesi, Antoine Gansel, Gereon Koßmann, Rene Schwonnek

quant-phSubmitted Aug 24, 2026• Announced Aug 25, 2026• 0 citations• Revised Sep 17, 2026 (v2)
v1 · Aug 24, 2026v2 · Sep 17, 2026
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Abstract

Translating observed data into a reliable estimate of the secure key rate is a crucial step for operating a quantum key distribution device. We provide a computational method for this task that only requires eigenvalue computations and is therefore both fast and resource efficient. In contrast, existing approaches rely on semidefinite programming or programming on the entropy cone, whose memory requirements can scale as $d^4$ in the underlying Hilbert-space dimension. Our method reduces this requirement to $d^2$. A minimal implementation of our algorithm takes fewer than 100 lines of Common Lisp. We demonstrate real-time key-rate estimation on a Raspberry Pi with a 1 GB memory and a Cortex-A53 processor. Despite these modest resources, our implementation outperforms existing workstation-based benchmarks by several orders of magnitude. Non-numerical verification can be incorporated with little overhead using rational approximations. These results open the way toward embedding complete numerical security analysis directly into qkd hardware.

Comments
16+18 pages, 8 figures; the updated version includes rational approximations

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Translating observed data into a reliable estimate of the secure key rate is a crucial step for operating a quantum key distribution device. We provide a computational method for this task that only requires eigenvalue computations and is therefore both fast and resource efficient. In contrast, existing approaches rely on semidefinite programming or programming on the entropy cone, whose memory requirements can scale as d4d^4d4 in the underlying Hilbert-space dimension. Our method reduces this requirement to d2d^2d2. A minimal implementation of our algorithm takes fewer than 100 lines of Common Lisp. We demonstrate real-time key-rate estimation on a Raspberry Pi with a 1 GB memory and a Cortex-A53 processor. Despite these modest resources, our implementation outperforms existing workstation-based benchmarks by several orders of magnitude. Non-numerical verification can be incorporated with little overhead using rational approximations. These results open the way toward embedding complete numerical security analysis directly into qkd hardware.
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.
Abstract Numerical security proofs offer a versatile approach for evaluating the secret-key generation rate of quantum key distribution (QKD) protocols. However, existing methods typically require perfect source characterization, which is unrealistic in practice due to the presence of inevitable encoding imperfections and side channels. In this paper, we introduce a novel security proof technique based on semidefinite programming that can evaluate the secret-key rate for both prepare-and-measure and measurement-device-independent QKD protocols when only partial information about the emitted states is available, significantly improving the applicability and practical relevance compared to existing numerical techniques. We demonstrate that our method can outperform current analytical approaches addressing partial state characterization in terms of achievable secret-key rates, particularly for protocols with non-qubit encoding spaces. This represents a significant step towards bridging the gap between theoretical security proofs and practical QKD implementations.
Security proof methods for quantum key distribution, QKD, that are based on the numerical key rate calculation problem, are powerful in principle. However, the practicality of the methods are limited by computational resources and the efficiency and accuracy of the underlying algorithms for convex optimization. We derive a stable reformulation of the convex nonlinear semidefinite programming, SDP, model for the key rate calculation problems. We use this to develop an efficient, accurate algorithm. The stable reformulation is based on novel forms of facial reduction, FR, for both the linear constraints and nonlinear quantum relative entropy objective function. This allows for a Gauss-Newton type interior-point approach that avoids the need for perturbations to obtain strict feasibility, a technique currently used in the literature. The result is high accuracy solutions with theoretically proven lower bounds for the original QKD from the FR stable reformulation. This provides novel contributions for FR for general SDP. We report on empirical results that dramatically improve on speed and accuracy, as well as solving previously intractable problems.
The use of decoy states in quantum key distribution (QKD) has provided a method for substantially increasing the secret key rate and distance that can be covered by QKD protocols with practical signals. The security analysis of these schemes, however, leaves open the possibility that the development of better proof techniques or better classical postprocessing methods might further improve their performance in realistic scenarios. In this paper, we derive upper bounds on the secure key rate for decoy-state QKD. These bounds are based basically only on the classical correlations established by the legitimate users during the quantum communication phase of the protocol. The only assumption about the possible postprocessing methods is that double click events are randomly assigned to single click events. Further, we consider only secure key rates based on the uncalibrated device scenario which assigns imperfections such as detection inefficiency to the eavesdropper. Our analysis relies on two preconditions for secure two-way and one-way QKD. The legitimate users need to prove that there exists no separable state (in the case of two-way QKD) or that there exists no quantum state having a symmetric extension (one-way QKD) that is compatible with the available measurements results. Both criteria have been previously applied to evaluate single-photon implementations of QKD. Here we use them to investigate a realistic source of weak coherent pulses. The resulting upper bounds can be formulated as a convex optimization problem known as a semidefinite program which can be efficiently solved. For the standard four-state QKD protocol, they are quite close to known lower bounds, thus showing that there are clear limits to the further improvement of classical postprocessing techniques in decoy-state QKD.
Abstract Device-independent quantum key distribution (DIQKD) provides the strongest form of secure key exchange, using only the input–output statistics of the devices to achieve information-theoretic security. Although the basic security principles of DIQKD are now well understood, it remains a technical challenge to derive reliable and robust security bounds for advanced DIQKD protocols that go beyond the previous results based on violations of the CHSH inequality. In this work, we present a framework based on semidefinite programming that gives reliable lower bounds on the asymptotic secret key rate of any QKD protocol using untrusted devices. In particular, our method can in principle be utilized to find achievable secret key rates for any DIQKD protocol, based on the full input–output probability distribution or any choice of Bell inequality. Our method also extends to other DI cryptographic tasks.
Device-independent quantum key distribution (DIQKD) is arguably the gold standard for secure quantum communication, as it aims to rely only on observed input-output statistics of an uncharacterized device which is only assumption to obey the laws of quantum physics. A corresponding security analysis hence demands a description of a quantum experiment from a most general perspective. Under close inspection, existing proof techniques do not always meet this goal as they tend to rely on subtile assumptions on a tensor product structure of the underlying Hilbert space and a 'hidden but finite' dimensionality. In this work, we collect the tools needed for a full analysis of DIQKD in the commuting operator framework, which avoids these subtilities and provides the arguably most general view on a quantum experiment. We rigorously proof the common assumption that in DIQKD measurements can be w.l.o.g. assumed to be projective. Furthermore, we show that task of computing key rates can be casted as a non-commutative polynomial optimization (NPO) problem to which the Navascués-Pironio-Acín (NPA) hierarchy gives a correct and converging relaxation. As a tool, we generalize the integral representation for the relative entropy by Frenkel [Quantum 7, 1102 (2023)] to general von Neumann algebras and apply techniques from Kossmann and Schwonnek [arXiv: 2411.04858] for the approximation in an NPO program.