Discrete-Modulated (DM) Continuous-Variable (CV) Quantum Key Distribution (QKD) addresses a long-standing challenge in quantum cryptography. It is an experimentally attractive approach offering high key rates over metropolitan-scale distances while utilising state-of-the-art telecom infrastructure. For more than two decades DM CV-QKD lacked a complete composable finite-size security proof against coherent attacks, despite its promise. Existing works either restricted the adversary or relied on unrealistic assumptions regarding side channel information.

The first thorough composable finite-size security analysis of DM CV-QKD under realistic conditions including coherent attacks has been completed. This closes a key gap in theoretical understanding and enables truly secure implementations of this promising QKD protocol. Researchers developed new analytical tools and refined existing techniques from quantum cryptography to rigorously bound the information leakage to an eavesdropper. A specific contribution is a tight, practical security bound applicable to currently deployed systems utilising 4096 dimensions with a modulation variance of 1.

Extended range and rigorous security validation for discrete modulated continuous variable QKD

Positive key rates exceeding 70km now demonstrate success in Discrete-Modulated Continuous-Variable Quantum Key Distribution (DM CV-QKD), representing a substantial advance over previous limitations yielding no secure keys beyond approximately 25km under comparable conditions. The University of Waterloo team and collaborating institutions achieved this through development of new theoretical tools including an infinite-dimensional marginal-constrained entropy accumulation theorem simplifying complex calculations, alongside a dimension reduction technique enabling practical application to currently deployed systems utilising up to 4096 dimensions.

The iMEAT (infinite-dimensional marginal-constrained entropy accumulation theorem) simplifies calculations involving complex quantum states without limiting their dimensionality. Furthermore, a dimension reduction technique allowed security bounds to be applied effectively even when employing high dimensional signalling up to 4096 dimensions streamlining analysis for real-world implementations. They also successfully established complete composable finite-size security proofs against coherent attacks incorporating imperfect detectors and both fixed and variable length protocol variants. The resulting key rates recover known asymptotic values and outperform existing finite-size bounds based on collective attacks offering positive key rates beyond 70km with experimentally relevant block sizes.

Composable Finite-Size Security Proofs for Discrete Modulation Continuous Variable QKD

Scientists have long sought complete composable finite-size security proofs for Discrete Modulation Continuous-Variable Quantum Key Distribution (DM CV-QKD) protocols against coherent attacks, incorporating imperfect detectors and both fixed- and variable-length variants. Establishing rigorous security guarantees for DM CV-QKD has progressed along several complementary directions; early works demonstrated feasibility under restricted attack scenarios or specific configurations while approximation techniques exploited that sufficiently large constellations can closely reproduce statistical properties of Gaussian modulation enabling security arguments for arbitrary schemes. A major breakthrough came with numerical security proofs based on semidefinite programming (SDP), which directly optimise over quantum states compatible with observed measurement data making them well suited to finite constellations.

Original SDP formulations relied on a photon-number cutoff assumption but continuity-bound dimension reduction later removed this restriction and enabled composable finite-size security against collective attacks and Gaussian attacks. Extending these approaches to general attacks remained difficult however, as existing methods either reintroduce cutoffs or fail to recover asymptotic key rates in the limit of large block sizes.

One attempt addressed fixed-length protocols using conic reformulation, yet its proof remains incomplete due to an assumption about Eve’s purification being finite dimensional. Modern communication systems derive much efficiency from one simple idea: information is conveyed using carefully designed finite signal alphabets.

Whether realised as quadrature-amplitude modulation in optical fibres or phase-shift keying often used in wireless links, constellations represent the interface between digital information and the physical quantity used to transmit a signal. Quantum Key Distribution (QKD) faces a similar design question; instead of transmitting classical bits alone, legitimate users Alice and Bob distribute and measure quantum states whose distinguishability fundamentally limits the information available to an eavesdropper Eve.

Consequently, the choice of signal alphabet directly influences both practicality and security. Continuous-Variable (CV) QKD protocols realise this alphabet through optical states measured by heterodyne detection making them uniquely compatible with modern optical communication hardware.

Historically these states have been sampled from Gaussian distributions because inherent symmetry provides powerful theoretical tools for security analyses. From a practical perspective however, communication hardware is designed for finite constellations rather than continuous distributions. Discrete-Modulation therefore promises a considerably more practical realisation of CV-QKD while preserving its compatibility with conventional telecom technology.

Replacing continuous modulation with a finite alphabet fundamentally alters the mathematical structure of the protocol; arguments underpinning many security proofs no longer apply requiring new theoretical tools to establish rigorous security guarantees. This work fully closes this gap through novel developments starting with the Rényi leftover-hashing lemma which bounds trace distance between Alice’s/Bob’s shared state after protocol execution and a perfectly random key unknown to Eve, expressed as conditional Rényi entropy.

Calculating this entropic quantity requires constrained optimisation subject to experimental input, so an entropy accumulation theorem was proven reducing calculation of n-round conditional Rényi entropy to single round optimisation allowing accrued sideinformation to be infinite dimensional while guaranteeing a marginal on the n-round state.

Identified as iMEAT (infinite-dimensional, marginal-constrained), it reduces to an optimisation with infinite dimensionality. A dimension reduction method then bounds value of that infinite-dimensional optimisation by finite one plus correction term enabling bounding key rate using numerically implementable problems without unjustified cutoff assumptions; unlike earlier works needing separate testing procedures for unknown quantum state weight, this develops a method determining weight during key rate bound determination removing slackness and streamlining protocol design relying only on POVM measurements linking problem to experimental observations. This yields asymptotically tight lower bounds on composable secure key rates against coherent attacks converging towards asymptotic rates.

The paper is structured as follows: first introducing the general discrete modulated continuous variable QKD protocol analysed; next defining security framework in Section III; then presenting iMEAT (Theorem 70) in Section IV followed by security proof of fixed- and variable length variants in Section V. While proven bounds cannot yet be evaluated, dimension reduction argument proving rigorous finite dimensionality for optimisation problems is presented in Section VI completing DM CV-QKD protocols’ security proof.

Details about numerical implementation solving convex optimisation alongside simulation model illustrating findings follow in Section VII with plots showcasing achievable secure keyrates found in Section VIII concluding the work in IX while detailed derivations/proofs reside within Appendices. Discrete-Modulated Continuous-Variable Quantum Key Distribution offers high key rates over metropolitan distances utilising existing telecom infrastructure.

However, this approach lacked a complete composable finite-size security proof against coherent attacks for over two decades. Previous efforts either limited adversaries to collective attacks, imposed additional assumptions, applied only to specific modulation formats or failed to recover known asymptotic rates. This work resolves this longstanding problem by establishing the first such proof for Discrete-Modulated CV-QKD protocols incorporating imperfect detectors and both fixed- and variable-length protocol variants; it introduces an infinite-dimensional marginal-constrained entropy accumulation theorem alongside a rigorous dimension reduction technique enabling numerically tractable bounds on infinite-dimensional security.

Modern communication systems benefit from utilising carefully designed signal alphabets to convey information efficiently. Continuous-Variable QKD realises this alphabet through optical states measured by heterodyne detection uniquely compatible with modern hardware.

Historically these states have been sampled from Gaussian distributions due to inherent symmetry aiding theoretical analysis; however, hardware is designed for finite constellations suggesting Discrete Modulation offers a more practical realisation while maintaining compatibility with telecom technology. Early works demonstrated feasibility under restricted attacks or specific configurations.

Approximation techniques exploited large constellation similarity to Gaussian modulation enabling broader arguments. Extending these methods to general attacks proved difficult as existing approaches reintroduced cutoffs or failed to recover asymptotic rates; recent attempts at fixed length protocols faced incompleteness due to assumptions about Eve’s purification dimensionality.

Rényi entropy bounds refine security proofs for continuous-variable quantum cryptography

The team’s breakthrough addresses a longstanding need for verifiable security in practical quantum communication systems, yet detailed analysis reveals complexities within the approach itself. A central challenge lies in accurately bounding Rényi entropy, a measure of uncertainty key for quantifying information leakage to an eavesdropper when dealing with infinitely large quantum states. Previous attempts relied on approximations or assumptions about Eve’s capabilities that ultimately undermined the completeness of the proof regarding her purifying register’s dimensionality; acknowledging these criticisms concerning bounding Rényi entropy and applying marginal-constrained entropy accumulation theorems is important for transparency within the field. Researchers have demonstrably addressed previously unproven aspects of discrete modulated continuous-variable quantum key distribution security resolving a longstanding problem hindering practical implementation. Their new approach yields improved key rates over longer distances exceeding seventy kilometres with realistic parameters establishing a firm foundation for deploying more secure communication networks utilising existing fibre optic infrastructure.

The researchers established the first complete, composable finite-size security proof for Discrete-Modulated Continuous-Variable Quantum Key Distribution protocols against coherent attacks. This resolves a long-standing issue preventing rigorous verification of this promising method for securing communications using standard telecom infrastructure. The resulting key rates match known theoretical limits and achieve positive values beyond 70km under experimentally relevant conditions, demonstrating enhanced performance compared to previous bounds based on collective attacks. Their work introduces new tools, an infinite-dimensional entropy accumulation theorem and dimension reduction technique, that improve the accuracy of quantifying information leakage in quantum cryptography systems.

👉 More information
🗞 Unconditional Security of Discrete-Modulated CV-QKD from Infinite-Dimensional MEAT
✍️ Lars Kamin, Ian George, John Burniston and Florian Kanitschar
🧠 ArXiv: https://arxiv.org/abs/2609.16105

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