Universal locality of quantum thermal susceptibility

The ultimate precision of any measurement of the temperature of a quantum system is the inverse of the local quantum thermal susceptibility [A. De Pasquale et al., Nat. Commun. 7, 12782 (2016)] of the subsystem with which the thermometer interacts. If this subsystem can be described with the canonical ensemble, such quantity reduces to the variance of the local Hamiltonian, which is proportional to the heat capacity of the subsystem. However, the canonical ensemble might not apply in the presence of interactions between the subsystem and the rest of the system. In this work, we address this problem in the framework of locally interacting quantum systems. We prove that the local quantum thermal susceptibility of any subsystem 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. This result greatly simplifies the determination of the ultimate precision of any local estimate of the temperature and rigorously determines the regime where interactions can affect this precision.

Figure 1

A 2D square lattice with two-site first-neighbor interactions. In red are the sites of $A$ and the edges of $\mathcal{A}$. In green are the sites $x$ with $d(x,A)=1$ and the edges of ${\mathcal{C}}_1$. The boundary edges of $A$ are the green edges connecting a red with a green site. In black are all the other sites and edges. Here, $N_{\partial }=6$.