We introduce an improved one-shot characterisation of randomness extraction against quantum side information (privacy amplification), strengthening known one-shot bounds and providing a unified derivation of the tightest known asymptotic constraints. Our main tool is a new class of smooth conditional entropies defined by lifting classical smooth divergences through measurements. For the key case of measured smooth Rényi divergence of order 2, we show that this can be alternatively understood as allowing for smoothing over not only states, but also non-positive Hermitian operators. Building on this, we establish a tightened leftover hash lemma, significantly improving over all known smooth min-entropy bounds on quantum privacy amplification and recovering the sharpest classical achievability results. We extend these methods to decoupling, the coherent analogue of randomness extraction, obtaining a corresponding improved one-shot bound. Relaxing our smooth entropy bounds leads to one-shot achievability results in terms of measured Rényi divergences, which in the asymptotic i.i.d. limit recover the state-of-the-art error exponent of [Dupuis, arXiv:2105.05342]. We show an approximate optimality of our results by giving a matching one-shot converse bound up to additive logarithmic terms. This yields an optimal second-order asymptotic expansion of privacy amplification under trace distance, establishing a significantly tighter one-shot achievability result than previously shown in [Shen et al., arXiv:2202.11590] and proving its optimality for all hash functions.

翻译：本文提出了一种改进的单次随机性提取（隐私放大）抗量子边信息的特征刻画方法，强化了已知的单次界，并为已知最紧致的渐近约束提供了统一推导。我们的主要工具是通过测量提升经典平滑散度而定义的一类新型平滑条件熵。针对关键的二阶测量平滑Rényi散度情形，我们证明其可等价理解为允许不仅对态进行平滑化，也可对非正定厄米算子进行平滑化。在此基础上，我们建立了强化的剩余哈希引理，显著改进了所有已知的量子隐私放大平滑最小熵界，并恢复了最尖锐的经典可达性结果。我们将这些方法推广到解耦——随机性提取的相干模拟，获得了相应的改进单次界。通过松弛我们的平滑熵界，我们得到了基于测量Rényi散度的单次可达性结果，其在渐近独立同分布极限下恢复了[Dupuis, arXiv:2105.05342]中的最优误差指数。我们通过给出匹配的单次逆界（至对数加性项）证明了结果的近似最优性。这导出了迹距离下隐私放大的最优二阶渐近展开，建立了比[Shen et al., arXiv:2202.11590]中先前结果显著更紧致的单次可达性结果，并证明了其对所有哈希函数的最优性。