Stochastic Quantum Information Geometry and Speed Limits at the Trajectory Level

Abstract: Standard quantum metrology relies on ensemble-averaged quantities, such as the Quantum Fisher Information (QFI), which often mask the fluctuations inherent to single-shot realizations. In this work, we bridge the gap between quantum information geometry and stochastic thermodynamics by introducing the Conditional Quantum Fisher Information (CQFI). Defined via the Symmetric Logarithmic Derivative, the CQFI generalizes the classical stochastic Fisher information to the quantum domain. We demonstrate that the CQFI admits a decomposition into incoherent (population) and coherent (basis rotation) contributions, augmented by a transient interference cross-term absent at the ensemble level. Crucially, we show that this cross-term can be negative, signaling destructive interference between classical and quantum information channels along individual trajectories. Leveraging this framework, we construct a stochastic information geometry that defines thermodynamic length and action for single quantum trajectories. Finally, we derive fundamental quantum speed limits valid at the single-trajectory level and validate our results using the quantum jump unraveling of a driven thermal qubit.