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Finite-Key Analysis of Quantum Key Distribution with Characterized Devices Using Entropy Accumulation

Ian George, Jie Lin, Thomas van Himbeeck, Kun Fang, Norbert Lütkenhaus

TL;DR

The paper introduces a finite-key framework for entanglement-based device-dependent QKD by adapting the Entropy Accumulation Theorem (EAT) to characterize entropy growth across sequential rounds with characterized devices. It furnishes practical tools: (i) sufficient, verifiable Markov-chain conditions for announcements, and (ii) two numerical algorithms to construct (near-)optimal min-tradeoff functions, the second incorporating second-order terms via Fenchel duality. By employing privacy amplification without smoothing and sandwiched Rényi entropies, the authors derive a tight, non-smoothed key-length bound that improves finite-key performance over traditional smooth-entropy bounds. The framework is demonstrated on several protocols (BB84, six-state, high-dimensional BB84-like, and SPDC-based BB84), revealing finite-size gains and highlighting regimes where EAT outperforms postselection approaches, especially in high dimension. Overall, the work provides a practical route to tighter finite-key rates for device-dependent QKD and lays foundational methods for broader applicability of Rényi-entropy-based security proofs in quantum cryptography.

Abstract

The Entropy Accumulation Theorem (EAT) was introduced to significantly improve the finite-size rates for device-independent quantum information processing tasks such as device-independent quantum key distribution (QKD). A natural question would be whether it also improves the rates for device-dependent QKD. In this work, we provide an affirmative answer to this question. We present new tools for applying the EAT in the device-dependent setting. We present sufficient conditions for the Markov chain conditions to hold as well as general algorithms for constructing the needed min-tradeoff function. Utilizing Dupuis' recent privacy amplification without smoothing result, we improve the key rate by optimizing the sandwiched Rényi entropy directly rather than considering the traditional smooth min-entropy. We exemplify these new tools by considering several examples including the BB84 protocol with the qubit-based version and with a realistic parametric downconversion source, the six-state four-state protocol and a high-dimensional analog of the BB84 protocol.

Finite-Key Analysis of Quantum Key Distribution with Characterized Devices Using Entropy Accumulation

TL;DR

The paper introduces a finite-key framework for entanglement-based device-dependent QKD by adapting the Entropy Accumulation Theorem (EAT) to characterize entropy growth across sequential rounds with characterized devices. It furnishes practical tools: (i) sufficient, verifiable Markov-chain conditions for announcements, and (ii) two numerical algorithms to construct (near-)optimal min-tradeoff functions, the second incorporating second-order terms via Fenchel duality. By employing privacy amplification without smoothing and sandwiched Rényi entropies, the authors derive a tight, non-smoothed key-length bound that improves finite-key performance over traditional smooth-entropy bounds. The framework is demonstrated on several protocols (BB84, six-state, high-dimensional BB84-like, and SPDC-based BB84), revealing finite-size gains and highlighting regimes where EAT outperforms postselection approaches, especially in high dimension. Overall, the work provides a practical route to tighter finite-key rates for device-dependent QKD and lays foundational methods for broader applicability of Rényi-entropy-based security proofs in quantum cryptography.

Abstract

The Entropy Accumulation Theorem (EAT) was introduced to significantly improve the finite-size rates for device-independent quantum information processing tasks such as device-independent quantum key distribution (QKD). A natural question would be whether it also improves the rates for device-dependent QKD. In this work, we provide an affirmative answer to this question. We present new tools for applying the EAT in the device-dependent setting. We present sufficient conditions for the Markov chain conditions to hold as well as general algorithms for constructing the needed min-tradeoff function. Utilizing Dupuis' recent privacy amplification without smoothing result, we improve the key rate by optimizing the sandwiched Rényi entropy directly rather than considering the traditional smooth min-entropy. We exemplify these new tools by considering several examples including the BB84 protocol with the qubit-based version and with a realistic parametric downconversion source, the six-state four-state protocol and a high-dimensional analog of the BB84 protocol.
Paper Structure (57 sections, 34 theorems, 174 equations, 8 figures)

This paper contains 57 sections, 34 theorems, 174 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation
  4. QKD security definition
  5. The Entropy Accumulation Theorem
  6. Formal statement
  7. Applying EAT to device-dependent QKD
  8. Tensor product structure
  9. Challenge 1: constructing optimal min-tradeoff functions
  10. Challenge 2: guaranteeing Markov chain conditions
  11. The Quantum Key Distribution Protocol
  12. Physical and virtual protocol description
  13. Assumptions on public announcements
  14. Example (optical discrete-variable protocols):
  15. Completeness of a QKD Protocol
  16. ...and 42 more sections

Key Result

Theorem 1

Consider EAT channels $\mathcal{M}_{1},...,\mathcal{M}_{n}$ and their output $\rho_{S^{n}_{1}P^{n}_{1}X^{n}_{1}E}$ such that it satisfies the Markov conditions and $S_{i}$ is a classical register for all $i \in [n]$. Let $h \in \mathbb{R}$, $\alpha \in (1,2)$, and $f$ be a min-tradeoff function for holds for where and $d_S = \max_{i\in [n]} |S_i|$ is the maximum dimension of the systems $S_i$.

Figures (8)

  • Figure 1: Diagrammatic depiction of EAT process and theorem. (a) Captures the overall process and (b) is the process captured by the min-tradeoff function (See \ref{['defn:mintradeofffunction']}). They are related by the Entropy Accumulation Theorem (\ref{['thm:EATv2']}). Note that $\mathcal{M}_{n}$ may be viewed as outputting a trivial register, $R_{n} \cong \mathbb{C}$, which we have suppressed. The output of the EAT process is $\rho_{S^{n}_{1}P^{n}_{1}X^{n}_{1}E}$, the output registers along with the purification.
  • Figure 2: Key rate versus the number of signals for the qubit BB84 protocol to compare two algorithms for the generation of min-tradeoff functions. The quantum bit error rate is set to $Q = 0.01$ in the simulation, and $\zeta_{t} = 0.005$ for defining the acceptance set $\mathcal{Q}$. The red circle marker corresponds to \ref{['alg:Algorithm1_min_tradeoff_function']} while the green star marker corresponds to \ref{['alg:Algorithm2_min_tradeoff_function']}. The key rate formula is based on \ref{['thm:keylengthwithoutsmoothing']}. Other protocol parameters are optimized as described in the main text.
  • Figure 3: Key rate versus the number of signals for the qubit BB84 protocol to compare different second-order correction terms using \ref{['alg:Algorithm2_min_tradeoff_function']}. The quantum bit error rate is set to $Q = 0.01$ in the simulation, and $\zeta_{t} = 0.005$ for defining the acceptance set $\mathcal{Q}$. The blue circle marker corresponds to the key rate formula given in \ref{['thm:keylengthwithoutsmoothing']}Dupuis2021 while the magenta star marker corresponds to the key rate formula from \ref{['thm:keyLengthWithSmoothing']}Dupuis2019.
  • Figure 4: (a) Key rate versus the number of signals for the qubit BB84 protocol for different 'sizes' of acceptance set controlled by parameter $\zeta_{t}$. (b) The fraction of the key rate per signal (KPS) at $\zeta_{t} = 0$ achieved for other values of $\zeta_{t}$. In both subplots, the quantum bit error rate is set to $Q = 0.01$ in the simulation. Other protocol parameters as described in the main text.
  • Figure 5: Key rate versus the number of signals for the six-state four-state protocol with two algorithms. The quantum bit error rate is set to $Q = 0.01$ in the simulation, and $\zeta_{t} = 0.005$ for defining the acceptance set $\mathcal{Q}$. This plot is obtained by using the key rate formula given in \ref{['thm:keylengthwithoutsmoothing']}Dupuis2021. The red circle marker corresponds to the min-tradeoff function construction by \ref{['alg:Algorithm1_min_tradeoff_function']} while the green star marker corresponds to the min-tradeoff function construction by \ref{['alg:Algorithm2_min_tradeoff_function']}.
  • ...and 3 more figures

Theorems & Definitions (88)

  • Definition 1: EAT Channels
  • Definition 2: Min-tradeoff functions
  • Remark 1
  • Theorem 1: Special Case of Proposition V.3 of Dupuis2019
  • Remark 2
  • Definition 3
  • Theorem 2
  • Remark 3
  • Definition 4
  • Proposition 3
  • ...and 78 more