Optimizing the Mathematics of QKD Key Generation for Faster Computation

“Forty years ago, not even Bennett and Brassard could have guessed how quantum cryptography would look today,”

says Dr. Carlos Pascual-García, Head of Quantum Theory at Luxquanta.

“The future is uncertain, and only time will tell the impact our study will have. But I am happy to share that we are already pursuing these broader goals, and preliminary results look really promising.”

When Charles Bennett and Gilles Brassard proposed the first quantum key distribution (QKD) protocol in 1984, the premise was groundbreaking: two people could share a secret key whose safety rests on the laws of physics themselves, not on hard-to-solve mathematical algorithms. More than four decades later, that promise has been refined into a mature theoretical, and practical, framework.

Behind the generation and distribution of QKD-keys, the actual rate at which a real Quantum Key Distribution device can produce secure key, still comes down to solving a demanding optimization problem for every protocol, every time.

At the center of that problem sits a quantity called Rényi entropy. This is a measure, borrowed from information theory, of how much uncertainty a worst-case eavesdropper is left with once all observed statistics are accounted for. Rényi entropies give tighter, more favorable security bounds than the older von Neumann entropy, especially when working with a realistic, finite number of signals rather than an idealized infinite stream. The catch is that computing them exactly has historically been slow, numerically fragile, and workable only for a narrow set of protocols.

A team including Mariana Navarro, carrying out a joint Industrial PhD thesis with Luxquanta and ICFO and led by Dr. Carlos Pascual-García, Head of Quantum Theory at Luxquanta and former ICFO researcher, together with Andrés González Lorente, Pablo Parellada, and Dr. Mateus Araújo from the University of Valladolid, has now closed that gap.

Published in PRX Quantum, the paper shows how to reformulate the Rényi-entropy calculation as a conic optimization problem. In other words, instead of approximating the curved entropy function step by step, they built a custom mathematical shape (a “cone”) whose geometry matches the entropy function exactly, including a careful treatment of the tricky edge cases where earlier methods tended to lose precision or stall.

The payoff is largely speed and reliability, with the same security guarantees, computed far more efficiently. In their benchmarks, calculations for the classic BB84 protocol that previously took tens of seconds now complete in a fraction of a second, and protocols that are harder to certify, such as Continuous-Variable QKD with a small set of coherent states, converge reliably where older generic solvers sometimes failed to converge at all.

“At first, we were amazed by the speed of our strategy,”

recalls Mariana Navarro, first author of the article.

“Moreover, our method is not only faster, but also more systematic and numerically precise than previous approaches,”

she adds.

Beyond raw speed, the authors point to a broader payoff: a fast, dependable way to compute these bounds makes it practical to explore questions that were previously too costly to study exhaustively, comparing alternative protocol designs on equal footing, quantifying how channel loss or detector imperfections erode the key rate, and mapping out how far apart two parties can be while a protocol remains provably secure.

The team is already extending the approach toward more complex, deployment-relevant protocols, including decoy-state BB84 and measurement-device-independent QKD.

Reference

Navarro, M., et al., Finite-size quantum key distribution rates from Rényi entropies using conic optimization, PRX Quantum 7, 033051 (2026). DOI: https://doi.org/10.1103/bf9s-m4jb

Acknowledgements:

MN and CPG were supported by the Government of Spain (Severo Ochoa CEX2019-000910-S, FUNQIP and NextGeneration EU PRTR-C17.I1) and European Union (QSNP, 101114043). MN acknowledges funding from the European Union’s Horizon Europe research and innovation programme under the MSCA Grant Agreement No. 101081441. CPG has received funding from the European Union’s Digital Europe Programme under the project QUARTER (101091588) and Horizon 2020 Research and Innovation Programme under the project QSNP (101114043), and from the European Innovation Council’s Horizon Europe EIC Accelerator Programme under the project MIQRO (101161539). The research of AGL, PVP, and MA was supported by the Q-CAYLE project, funded by the European Union-Next Generation UE/MICIU/Plan de Recuperación, Transformación y Resiliencia/Junta de Castilla y León (PRTRC17.11), and also by the Department of Education of the Junta de Castilla y León and FEDER Funds (Reference: CLU-2023-1-05). PVP has also been funded under the UVa 2024 predoctoral contract, co-financed by Banco Santander. MA was also supported by the Spanish Agencia Estatal de Investigación, Grant Nos.~RYC2023-044074-I and PID2024-161725OA-I00.