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

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.