Surpassing the thermal Cramér-Rao bound with collisional thermometry

In collisional thermometry, a system in contact with the thermal bath is probed by a stream of ancillas. Coherences and collective measurements were shown to improve the Fisher information in some parameter regimes, for a stream of independent and identically prepared (i.i.d.) ancillas in some specific states [Seah et al., Phys. Rev. Lett. 123, 180602 (2019)]. Here we refine the analysis of this metrological advantage by optimizing over the possible input ancilla states, also for block-i.i.d. states of block size $b=2$. For both an indirect measurement interaction and a coherent energy exchange channel, we show when the thermal Cram\'er-Rao bound can be beaten, and when a collective measurement of $N>1$ ancilla may return advantages over single-copy measurements.