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arXiv:2605.06249

Finite-size general security for relativistic phase shift keying via variable-length quantum key distribution

Carlos Pascual-García

quant-phSubmitted May 7, 2026• Announced May 8, 2026• 0 citations• Revised Jun 30, 2026 (v2)
v1 · May 7, 2026v2 · Jun 30, 2026
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Abstract

Differential phase shift keying constitutes a pathway towards practical quantum key distribution by using affordable commercial technologies, and robust theoretical foundations. Recent advances have proven its security against general adversaries, albeit requiring limitations, including strong repetition rate constraints at the security proof and costly statistical estimators. In this work, we overcome said limitations by considering an alternative scheme, herewith denominated relativistic phase shift keying (RPSK). We leverage variable-length general security techniques via entropy accumulation, together with methods based on Rényi leftover hashing and conic optimization. Our approach achieves secret key rates with $10^5$ signals beyond 12 dB, constituting a robust proof of the experimental implementability of RPSK.

Comments
18 pages, 4 figures. Former claims about DPSK have been removed

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Differential phase shift keying constitutes a pathway towards practical quantum key distribution by using affordable commercial technologies, and robust theoretical foundations. Recent advances have proven its security against general adversaries, albeit requiring limitations, including strong repetition rate constraints at the security proof and costly statistical estimators. In this work, we overcome said limitations by considering an alternative scheme, herewith denominated relativistic phase shift keying (RPSK). We leverage variable-length general security techniques via entropy accumulation, together with methods based on Rényi leftover hashing and conic optimization. Our approach achieves secret key rates with 10510^5105 signals beyond 12 dB, constituting a robust proof of the experimental implementability of RPSK.
The design of quantum protocols for secure key generation poses many challenges: On the one hand, they need to be practical concerning experimental realisations. On the other hand, their theoretical description must be simple enough to allow for a security proof against all possible attacks. Often, these two requirements are in conflict with each other, and the differential phase shift (DPS) QKD protocol exemplifies these difficulties: It is designed to be implementable with current optical telecommunication technology, which, for this protocol, comes at the cost that many standard security proof techniques do not apply to it. After about 20 years since its invention, this work presents the first full security proof of DPS QKD against general attacks, including finite-size effects. The proof combines techniques from quantum information theory, quantum optics, and relativity. We first give a security proof of a QKD protocol whose security stems from relativistic constraints. We then show that security of DPS QKD can be reduced to security of the relativistic protocol. In addition, we show that coherent attacks on the DPS protocol are, in fact, stronger than collective attacks. Our results have broad implications for the development of secure and reliable quantum communication technologies, as they shed light on the range of applicability of state-of-the-art security proof techniques.
Differential-phase-shift (DPS) quantum key distribution (QKD) is one of the major QKD protocols that can be implemented with a simple setup using a laser source and a passive detection unit. Recently, an information-theoretic security proof of this protocol was established by Mizutani et al. [npj Quantum Inf. 5, 87 (2019)], assuming an infinitely large number of emitted pulses. To implement the DPS protocol in a real-life world, it is indispensable to analyze the security with the finite number of emitted pulses. The extension of the security proof to the finite-size regime requires the accommodation of the statistical fluctuations to determine the amount of privacy amplification. In doing so, Azuma's inequality is often employed, but unfortunately we show that in the case of the DPS protocol, this results in a substantially low key rate. This low key rate is due to a loose estimation of the sum of probabilities regarding three-photon emission whose probability of occurrence is very small. The main contribution of our work is to show that this obstacle can be overcome by exploiting the recently found novel concentration inequality, Kato's inequality. As a result, the key rate of the DPS protocol is drastically improved. For instance, assuming typical experimental parameters, a 3-Mbit secret key can be generated over 77 km for 8.3 hours, which shows the feasibility of DPS QKD under a realistic setup.
Continuous-variable quantum key distribution (CVQKD) is compatible with telecommunication infrastructure, but implementing composable security with experimentally practical resources has remained challenging, particularly for discrete-modulated (DM) protocols. We report the first experimental demonstration of a DM CVQKD system that generates composable secret keys against general attacks with finite-size block lengths on the order of ∼106\sim 10^{6}∼106 rounds via a quadrature phase shift keying (QPSK) system. Our implementation follows a variable-length, general security framework enabled by modern entropy accumulation techniques and conic optimization, whose experimental pipeline allows real-time operation on near-commercial hardware.
Among many quantum key distribution protocols, the round-robin differential-phase-shift (RRDPS) protocol is unique in that it can upper-bound the amount of information leakage without monitoring signal disturbance. To expedite implementation of the protocol, however, the number of pulses forming a single block should be kept small, which significantly decreases the key rates in the original security proof. In the present paper, we refine the security proof of the RRDPS protocol in the finite-sized regime and achieve a tighter estimation for the information leakage with neither monitoring signal disturbance nor changing the original experimental setups. As a consequence, we obtain better key rates in both asymptotic and finite-sized cases while keeping the preferable features of the protocol, such as omission of phase randomization.
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.
We develop a flexible and robust framework for finite-size security proofs of quantum key distribution (QKD) protocols under coherent attacks, applicable to both fixed- and variable-length protocols. Our approach achieves high finite-size key rates across a broad class of protocols while imposing minimal requirements. In particular, it eliminates the need for restrictive conditions such as limited repetition rates or the implementation of virtual tomography procedures. To achieve this goal, we introduce new numerical techniques for the evaluation of sandwiched conditional Rényi entropies. In doing so, we also find an alternative formulation of the "QKD cone" studied in previous work. We illustrate the versatility of our framework by applying it to several practically relevant protocols, including decoy-state protocols. Furthermore, we extend the analysis to accommodate realistic device imperfections, such as independent intensity and phase imperfections. Overall, our framework provides both greater scope of applicability and better key rates than existing techniques, especially for small block sizes, hence offering a scalable path toward secure quantum communication under realistic conditions