Probing parameters estimation with Gaussian non-commutative measurements

Gaussian quantum states and channels are pivotal across many branches of quantum science and their applications, including the processing and storage of quantum information, the investigation of thermodynamics in the quantum regime, and quantum computation. The great advantage is that Gaussian states are experimentally accessible via their first and second statistical moments. In this work, we investigate parameter estimation for Gaussian states, in which the probe-state preparation stage involves two noncommutative Gaussian measurements on the position and momentum observables, introducing tunable parameters. The influence of these noncommutative Gaussian measurements is investigated through the quantum Fisher information (QFI). We showed that the QFI for characterizing Gaussian channels can be increased by adjusting the uncertainty parameters in the preparation of the probe state. Furthermore, if the probe is initially in a thermal state, probe-state preparation may generate quantum coherence in its energy basis. We showed that not only does the amount of coherence affect the improvement of the QFI, but also the rate of change of the coherence with respect to the parameter to be estimated. The proposed probe-state protocol is applied to two paradigmatic single-mode Gaussian channels, the attenuator and amplification channels, which are building blocks of Gaussian quantum information. Our results contribute to the use of coherence in quantum metrology and are experimentally feasible in quantum-optical devices.