重新思考量子平滑熵：量子隐私放大的一 shot 分析
Rethinking quantum smooth entropies: Tight one-shot analysis of quantum privacy amplification
AI总结：
本文提出改进的量子隐私放大一 shot 随机性提取刻画，强化了已知的一 shot 边界并统一推导了最紧的渐近约束。主要工具是通过测量提升经典平滑发散的新类平滑条件熵。
摘要：
我们引入了一种改进的一 shot 随机性提取对抗量子侧信息（隐私放大）的刻画，强化了已知的一 shot 边界并提供了最紧渐近约束的统一推导。我们的主要工具是通过测量提升经典平滑发散的新类平滑条件熵。一个关键角色是由测量平滑的 Rényi 相对熵阶 2，我们证明其具有等价的变分形式：它可以被理解为允许对状态以及非正 Hermitean 算符进行平滑。基于此，我们建立了改进的 leftover hash 引理，显著改进了所有已知的平滑 min-entropy 对可提取随机性的界，并恢复了最紧的经典可实现结果。我们将这些方法扩展到解耦，即隐私放大的一致 analogue，获得了相应的改进一 shot 边界。放松我们的平滑熵边界导致了以测量 Rényi 发散表示的一 shot 可行性结果，收紧了 [Dupuis, arXiv:2105.05342] 的边界并恢复了其中展示的最先进渐近 i.i.d. 错误指数。我们通过给出一个匹配的一 shot 反向边界（至多加法对数项）来证明我们的结果的近似最优性。这导致了在迹距离下的隐私放大最优的渐近展开，建立了比 [Shen 等，arXiv:2202.11590] 中之前展示的更紧的一 shot 可行性结果，并证明了所有哈希函数的最优性。
英文摘要：
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, tightening the bounds of [Dupuis, arXiv:2105.05342] and recovering state-of-the-art asymptotic i.i.d. error exponents. 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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
