Metrology in the presence of thermodynamically consistent measurements

Thermodynamically consistent measurements can preserve either statistics (unbiased) or marginal states (noninvasive) but not both. Here we show the existence of metrological tasks which unequally favor each of the aforementioned measurement types. We consider two different metrology tasks, namely, the weak value amplification technique and repeated metrology without resetting. We observe that unbiased measurement is better than noninvasive measurement for the former and the converse is true for the latter. We provide finite-temperature simulations of transmon sensors which estimate how much cooling, a resource for realistic measurements, is required to perform these metrology tasks.