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. A key role is played by the measured smooth Rényi relative entropy of order 2, which we show to admit an equivalent variational form: it can be 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 extractable randomness and recovering the sharpest classical achievability results. We extend these methods to decoupling, the coherent analogue of privacy amplification, 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：我们针对量子边信息（隐私放大）引入了一种改进的随机性提取单次刻画方法，强化了已知的单次界，并为最紧致的已知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光滑散度而定义的一类新型光滑条件熵。其中，二阶测量光滑Rényi相对熵发挥着关键作用，我们证明其具有等价的变分形式：该熵可理解为不仅允许对量子态进行光滑化处理，还允许对非正定厄米算符进行光滑化。基于此，我们建立了强化的剩余哈希引理，显著改进了所有已知关于可提取随机性的光滑最小熵界，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——隐私放大的相干模拟，获得了相应的改进单次界。通过松弛我们的光滑熵界，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]中最先进的误差指数。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。