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Composable Finite-Size Security of High-Dimensional Quantum Key Distribution Protocols

Florian Kanitschar, Marcus Huber

TL;DR

The paper tackles the challenge of securing high-dimensional QKD over noisy free-space and satellite channels by developing the first composable finite-size security proof against both collective and coherent attacks for general HD-QKD with practical measurements. It introduces a semi-analytic method to bound key rates based on min-entropy, and extends the security to coherent attacks via the postselection technique, including a robust variable-length security framework that adapts to atmospheric fluctuations. Applying these methods to a $d=16$ temporal-entanglement HD-QKD protocol, the results show nonzero finite-size key rates at realistic block sizes and demonstrate substantial gains from variable-length security, especially under rapidly varying channel conditions. The work thus provides a scalable, practical framework for implementing HD-QKD in satellite and free-space environments and bridges the gap between theory and experiment for high-dimensional quantum cryptography.

Abstract

Practical implementations of Quantum Key Distribution (QKD) extending beyond urban areas commonly use satellite links. However, the transmission of quantum states through the Earth's atmosphere is highly susceptible to noise, restricting its application primarily to nighttime. High-dimensional (HD) QKD offers a promising solution to this limitation by employing high-dimensionally entangled quantum states. Although experimental platforms for HD QKD exist, previous security analyses have been limited to the asymptotic regime and have either relied on impractical measurements or employed computationally demanding convex optimization tasks restricting the security analysis to low dimensions. In this work, we bridge this gap by presenting a composable finite-size security proof against both collective and coherent attacks for a general HD QKD protocol that uses only experimentally accessible measurements. In addition to the conventional, yet impractical `one-shot' key rates, we provide a practical variable-length security argument that yields significantly higher expected key rates. This approach is particularly crucial under rapidly changing and turbulent atmospheric conditions, as encountered in free-space and satellite-based QKD platforms.

Composable Finite-Size Security of High-Dimensional Quantum Key Distribution Protocols

TL;DR

The paper tackles the challenge of securing high-dimensional QKD over noisy free-space and satellite channels by developing the first composable finite-size security proof against both collective and coherent attacks for general HD-QKD with practical measurements. It introduces a semi-analytic method to bound key rates based on min-entropy, and extends the security to coherent attacks via the postselection technique, including a robust variable-length security framework that adapts to atmospheric fluctuations. Applying these methods to a temporal-entanglement HD-QKD protocol, the results show nonzero finite-size key rates at realistic block sizes and demonstrate substantial gains from variable-length security, especially under rapidly varying channel conditions. The work thus provides a scalable, practical framework for implementing HD-QKD in satellite and free-space environments and bridges the gap between theory and experiment for high-dimensional quantum cryptography.

Abstract

Practical implementations of Quantum Key Distribution (QKD) extending beyond urban areas commonly use satellite links. However, the transmission of quantum states through the Earth's atmosphere is highly susceptible to noise, restricting its application primarily to nighttime. High-dimensional (HD) QKD offers a promising solution to this limitation by employing high-dimensionally entangled quantum states. Although experimental platforms for HD QKD exist, previous security analyses have been limited to the asymptotic regime and have either relied on impractical measurements or employed computationally demanding convex optimization tasks restricting the security analysis to low dimensions. In this work, we bridge this gap by presenting a composable finite-size security proof against both collective and coherent attacks for a general HD QKD protocol that uses only experimentally accessible measurements. In addition to the conventional, yet impractical `one-shot' key rates, we provide a practical variable-length security argument that yields significantly higher expected key rates. This approach is particularly crucial under rapidly changing and turbulent atmospheric conditions, as encountered in free-space and satellite-based QKD platforms.
Paper Structure (24 sections, 5 theorems, 55 equations, 7 figures)

This paper contains 24 sections, 5 theorems, 55 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Protocol
  3. Security Argument
  4. Security against i.i.d. collective attacks
  5. Lift to coherent attacks
  6. Variable-Length Security
  7. Security against collective attacks
  8. Security of variable-length HD-QKD protocols against coherent attacks
  9. Security proof method
  10. Key rate evaluation
  11. Formalization of the protocol
  12. Results
  13. Fixed-length key rates
  14. Variable-length key rates
  15. Discussion
  16. ...and 9 more sections

Key Result

Theorem 1

Let $\Theta$ be the set of Alice's and Bob's observables. Let $\mathbf{r} \in \mathbb{R}^{|\Theta|}$ and $\mathbf{t} \in \mathbb{R}^{|\Theta|}_{\geq 0}$, where $|\Theta|$ denotes the cardinality of $\Theta$ and choose $\epsilon_{\mathrm{AT}} > 0$. Define the set of accepted statistics as and the corresponding acceptance set as where $r_{X}$ is the $X$-th element of the vector $\mathbf{r}$ and li

Figures (7)

  • Figure 1: Flowchart illustrating the variable-length decision. The length of the key is now a function of the observed statistics $\vec{F}^{\mathrm{obs}}$.
  • Figure 2: Composable secure key rates against (a) i.i.d. collective attacks and (b) coherent attacks for different block sizes $N$ and constant splitting ratio of $1\%$. The system dimension was fixed to $d=16$ and we used the witness $\mathrm{KH1}$.
  • Figure 3: Composable secure key rates against i.i.d. collective attacks versus loss for different block sizes $N$, splitting ratios $r_s$ and noise levels, parametrized by the number of solar photons $n_{\mathrm{sol}}$. The system dimension was fixed to $d=16$.
  • Figure 4: Comparison of our results with asymptotic behaviour and numerical key rate curves according to Ref. BKPH_2023. The splitting ratio was fixed to $r_s = 20\%$ and the noise level, parametrized by the number of solar photons, was chosen to be $n_{\mathrm{sol}} = 10^2$. The system dimension varies and is given in the legend.
  • Figure 5: Variable-length key rates for the 'Statistically Fluctuating Channel'. The system dimension was fixed to $d=16$
  • ...and 2 more figures

Theorems & Definitions (5)

  • Theorem 1: Acceptance Test Kanitschar_2023
  • Theorem 2: Security statement for collective attacks
  • Theorem 3: Security statement for coherent attacks
  • Theorem 4: Variable-length security statement for collective attacks
  • Theorem 5: Variable-length security statement for coherent attacks