  Thermodynamically consistent measurements can either preserve statistics
(unbiased) or preserve marginal states (non-invasive) 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 weak value amplification technique and repeated metrology without
resetting. We observe that unbiased measurement is better than non-invasive
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.
