Metrology of open quantum systems from emitted radiation

We explore the task of learning about the dynamics of a Markovian open quantum system by monitoring the information it radiates into its environment. For an open system with Hilbert space dimension $D$, the quantum state of the emitted radiation can be described as a temporally ordered matrix-product state (MPS). We provide simple analytical expressions for the quantum Fisher information (QFI) of the radiation state, which asymptotically scales linearly with the sensing time unless the open system has multiple steady states. We characterize the crossovers in QFI near dynamical phase transitions, emphasizing the role of temporal correlations in setting the asymptotic rate at which QFI increases. We discuss when optimal sensing is possible with instantaneously measured radiation.