Time-local optimal control for parameter estimation in the Gaussian regime

Information about a classical parameter encoded in a quantum state can only decrease if the state undergoes a non-unitary evolution, arising from the interaction with an environment. However, instantaneous control unitaries may be used to mitigate the decrease of information caused by an open dynamics. A possible, locally optimal (in time) choice for such controls is the one that maximises the time-derivative of the quantum Fisher information (QFI) associated with a parameter encoded in an initial state. In this study, we focus on a single bosonic mode subject to a Markovian, thermal master equation, and determine analytically the optimal time-local control of the QFI for its initial squeezing angle (optical phase) and strength. We show that a single initial control operation is already optimal for such cases and quantitatively investigate situations where the optimal control is applied after the open dynamical evolution has begun.