Distributed source simulation is the task where two (or more) parties share some correlated randomness and use local operations and no communication to convert this into some target correlation. Wyner's seminal result showed that asymptotically the rate of uniform shared randomness needed for this task is given by a mutual information induced measure, now referred to as Wyner's common information. This asymptotic result was extended by Hayashi in the quantum setting to separable states, the largest class of states for which this task can be performed to vanishing error. In this work we characterize this task in a near-tight manner in the one-shot setting using the smooth entropy framework. We do this by introducing one-shot operational quantities and correlation measures that characterize them. We establish asymptotic equipartition properties for our correlation measures thereby recovering the previous vanishing-error asymptotic results. In doing so, we consider technical points in one-shot network information theory and provide methods for cardinality bounds in the smooth entropy calculus. We also introduce entangled state versions of the distributed source simulation task and determine bounds in this setting via quantum embezzling. This provides a strong characterization of this network task in the one-shot, quantum regime.

翻译：分布式源模拟任务中，两方（或多方）共享相关随机性，仅通过局部操作且无需通信，将其转换为目标相关性。Wyner的开创性结果表明，该任务渐近所需的均匀共享随机性速率由互信息诱导的度量给出，现称为Wyner公共信息。Hayashi在量子场景中将此渐近结果推广至可分态——这是该任务能以可忽略误差执行的态的最大类别。本研究利用平滑熵框架，在单次设定中以近乎紧致的界刻画该任务，通过引入单次操作量及相关性度量实现。我们建立了相关性度量的渐近等分性质，从而恢复了先前可忽略误差的渐近结果。在此过程中，我们探讨了单次网络信息论的技术要点，并提供了平滑熵演算中基数界的方法。此外，我们引入了分布式源模拟任务的纠缠态版本，并通过量子挪用技术确定了该设定下的界。这为单次量子体系下的网络任务提供了强有力的表征。