标题： 基于随机表示的经典和量子不确定性关系的统一框架 显示英文标题

标题： A unified framework for classical and quantum uncertainty relations using stochastic representations

摘要： 热力学不确定性关系（TURs）和动力学不确定性关系（KURs）提供了测量精度与热力学成本（如熵产生和活性）之间的权衡关系。传统上，这些关系是通过Cramér-Rao不等式推导出来的，该不等式涉及确定性微分方程中的一种辅助扰动，这些方程描述了系统概率分布的时间演化。在本研究中，我们不依赖于基于确定性演化方程的先前公式，证明了所有之前发现的不确定性关系可以通过相同动力学的随机表示单独推导出来。为此，我们提出了一种基于随机表示的统一方法，适用于一般的马尔可夫动力学。超越经典系统，我们通过展开其动力学将这种方法应用于马尔可夫开放量子系统，在量子效应起重要作用的区域，推导出物理上更易访问且更紧致的量子不确定性关系。这完全确立了不确定关系作为经典和量子系统随机性质的内在属性。 显示英文摘要

摘要： Thermodynamic uncertainty relations (TURs) and kinetic uncertainty relations (KURs) provide tradeoff relations between measurement precision and thermodynamic cost such as entropy production and activity. Conventionally, these relations are derived using the Cram\'er-Rao inequality, which involves an auxiliary perturbation in deterministic differential equations governing the time evolution of the system's probability distribution. In this study, without relying on the previous formulation based on deterministic evolving equation, we demonstrate that all previously discovered uncertainty relations can be derived solely through the stochastic representation of the same dynamics. For this purpose, we propose a unified method based on stochastic representations for general Markovian dynamics. Extending beyond classical systems, we apply this method to Markovian open quantum systems by unraveling their dynamics, deriving quantum uncertainty relations that are physically more accessible and tighter in regimes where quantum effects play a significant role. This fully establishes uncertainty relations for both classical and quantum systems as intrinsic properties of their stochastic nature.