Current technology permits the realization of extremely small thermometers, conceived in order to carry out the challenging task of controlling the thermodynamical behavior of physical systems at the spatial resolution of the micro- and nanometer length scales. In this regime the quantum
correlations shared among the subcomponents of the system can play a non-trivial role in the measurement of the system temperature. We have approached this kind of questions by feeding them into the framework of quantum estimation theory.
Notably, in a recent paper  we have introduced a kind of mesoscopic version of the heat capacity, the so-called local quantum thermal susceptibility, which rigorously quantifies the highest achievable accuracy for estimating the temperature of a quantum system at thermal equilibrium when technical/practical limitations restrict our capabilities to local probing. This analysis fills a gap left uncovered by the typical mindset underpinning quantum and classical thermodynamics statements, according to which the thermal properties of physical systems are retrieved from averaging procedures which smoothen out local details. Exploiting the basic concepts of quantum estimation theory, our method provides an operative strategy to address the local thermal response of arbitrary quantum systems at equilibrium, without introducing any additional hypothesis neither on the system Hamiltonians nor on the state of the local subsystems.
The computation of the local quantum thermal susceptibility requires an exact diagonalization of the density matrix associated to the probed subsystem. This in general represents a quite non trivial point to be addressed. However in  we manage to show that this functional is close to the variance of its local Hamiltonian, provided the volume to surface ratio of the subsystem is much larger than the correlation length. Apart from greatly simplyfing the determination of the ultimate precision of any local estimate of the temperature, this result also rigorously determines the regime where interactions can affect the local thermometric precision.
 A. De Pasquale, D. Rossini, R. Fazio and V. Giovannetti, Nat. Commun. 7, 12782 (2016)
 G. De Palma, A. De Pasquale and V. Giovannetti, arXiv:1611.05738