Finite Block Length Analysis on Quantum Coherence Distillation and Incoherent Randomness Extraction
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We introduce a variant of randomness extraction framework in the context of quantum coherence theory where free incoherent operations are allowed before the incoherent measurement and the randomness extractors. This cryptographic framework opens a new perspective to the study of quantum coherence distillation by an exact one-shot relation, that is, the maximum number of random bits extractable from a given quantum state is precisely equal to the maximum number of coherent bits that can be distilled from the same state. This relation enables us to derive tight second order expansions of both tasks in the independent and identically distributed setting. Remarkably, the incoherent operation classes that can empower coherence distillation for generic states all admit the same second order expansions, indicating their operational equivalence for coherence distillation in both asymptotic and large block length regimes. As a by-product, we showcase an alternative proof of the strong converse property of coherence distillation and randomness extraction from their second order expansions.
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