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.