Entropy Bounds via Hypothesis Testing and Its Applications to Two-Way Key Distillation in Quantum Cryptography

TL;DR

This work forges a rigorous link between two-way advantage distillation key rates in QKD and quantum hypothesis testing by exploiting an integral representation of the conditional entropy. It yields tighter finite-blocklength entropy bounds and clarifies asymptotic conditions for positive key rates, translating quantum state discrimination performance into secure-key guarantees. The authors provide practical computational methods that reduce quantum discrimination to classical problems, enabling bounds for blocklengths up to about $n\approx 1000$ and establishing a necessary-and-sufficient asymptotic criterion $Q(\rho_{E|00},\rho_{E|11}) > \epsilon/(1-\epsilon)$ in general, with a stronger result $\lim_{n\to\infty} \mathscr{R}_n = 0$ for pure qubit cases when the criterion fails. This work bridges QKD security analysis with quantum hypothesis testing, offering a flexible framework to tighten key-rate estimates and guide device-dependent and device-independent AD protocols.

Abstract

Quantum key distribution (QKD) achieves information-theoretic security, without relying on computational assumptions, by distributing quantum states. To establish secret bits, two honest parties exploit key distillation protocols over measurement outcomes resulting after the the distribution of quantum states. In this work, we establish a rigorous connection between the key rate achievable by applying two-way key distillation, such as advantage distillation, and quantum asymptotic hypothesis testing, via an integral representation of the relative entropy. This connection improves key rates at small to intermediate blocklengths relative to existing fidelity-based bounds and enables the computation of entropy bounds for intermediate to large blocklengths. Moreover, this connection allows one to close the gap between known sufficient and conjectured necessary conditions for key generation in the asymptotic regime, while the precise finite blocklegth conditions remain open. More broadly, our work shows how advances in quantum multiple hypothesis testing can directly sharpen the security analyses of QKD.

Figures (2)

• Figure 1: Different lower bounds on $H(C|E)_{\rho}$ as a function of the parameter $\lambda$, where $\rho = \frac{1}{2} \sum_{i} | i \rangle\!\langle i | \otimes \tau_i$. The state $\tau_0$ is fixed as the qubit with Bloch vector $(0, 0, 1)$, while $\tau_1$ has Bloch vector $\left( \lambda/\sqrt{2}, 0, \lambda/\sqrt{2 } \right)$ with $\lambda \in [0, 1]$. The integral bound is obtained using \ref{['eqn: intergral_main_text']} and subsequently bounding the error probabilities. The Fidelity bound is computed using \ref{['eqn: error_probability_fidelity']} and the Error probability bound corresponds to the one obtained via \ref{['eqn: error_probability_symmetry']}.
• Figure 2: Comparison of entropy lower bounds. Our method is benchmarked against the fidelity‑based bound obtained from Eq. \ref{['eqn: Fidelity_lower_bound']}. For this example, we take two two‑qubit states $\omega_{0}$ and $\omega_{1}$ with fidelity $F(\omega_0,\omega_1)= 0.684$.

Theorems & Definitions (44)

• Proposition 1
• Definition 1: Petz Rényi Quantity
• Definition 2: Sandwiched Rényi Quantity
• Definition 3: Rényi Divergences
• Theorem 1: Tan et. al. tan2020advantage, Bae et. al. bae2007key
• Lemma 1
• proof
• Theorem 2
• proof
• Lemma 2