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

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.

💡 Research Summary

This paper establishes a rigorous link between the secret‑key rate achievable by two‑way key‑distillation protocols—specifically advantage distillation (AD)—and quantum asymptotic hypothesis testing. By exploiting a recently derived integral representation of the quantum relative entropy, the authors express the conditional von Neumann entropy of the classical‑quantum (cq) states that arise after the quantum‑channel stage directly in terms of binary state‑discrimination error probabilities. Proposition 1 shows that for a cq‑state (\rho=\sum_{c}p_c|c\rangle!\langle c|\otimes\omega_c) the conditional entropy can be written as an integral over the error probabilities of the ensembles ({s\omega_0,(1-s)\omega_1}) and its complement. This formulation immediately reproduces known lower bounds based on fidelity (via the Fuchs–van de Graaf inequality) and on the min‑entropy, but also enables the construction of tighter bounds by taking point‑wise maxima of multiple error‑probability estimates (e.g., fidelity‑based and the (s=1/2) bound).

On the upper‑bound side, the Audenaert–Alberti–Uhlmann inequality is employed to bound the error probability from above, which after integration yields a family of upper bounds parameterised by (\alpha\in