  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.
