Finite Block Length Analysis on Quantum Coherence Distillation and Incoherent Randomness Extraction

We give the first systematic study on second order asymptotics of operational task coherence distillation with and without assistance. In unassisted setting, we introduce a variant randomness extraction framework where free incoherent operations are allowed before measurement extractors. then show that maximum number random bits extractable from given quantum state is precisely equal to coherent can be distilled same state. This relation enables us derive tight expansions both tasks in independent identically distributed setting. Remarkably, operation classes empower for generic states all admit expansions, indicating their equivalence asymptotic large block length regime. generalize above line research assisted arising naturally bipartite systems Bob distills at hand, aided by benevolent Alice possessing other system. More precisely, new establish an exact between this distillation. strengthens one-shot setting confirms cryptographic indeed offers perspective Likewise, yields characterizations tasks. As by-products, strong converse property aforementioned expansions.