Metrological advantage at finite temperature for Gaussian phase estimation

The use of pure classical probes in a metrological experiment gives precision limited by the standard quantum limit. This limit may be circumvented using nonclassical probes. Since the onset of quantum metrology, numerous papers have studied the standard quantum limit in various contexts. This allowed to better understand the resources needed to surpass this limit, achieving a metrological advantage. However, a general characterization of said resources is still missing. Moreover, the standard quantum limit itself can be unattainable at the classical level due to noise in the preparation procedure. Here, in the context of phase estimation, we introduce a quantifiable definition of metrological advantage that takes into account noise in the preparation procedure. We characterize all Gaussian states that possess this metrological advantage, and show that squeezing is not only necessary, but sufficient, to have an advantage. Finally, we compare our results to the framework of resource theory; interestingly, we find that some, but not all, properties of our metrological advantage can be recast in this language