Role of topology in determining the precision of a finite thermometer

TL;DR

This paper investigates how the topology of a finite system influences its ability to serve as a precise thermometer, using graph theory and Fisher information to analyze the effects of connectivity on temperature measurement accuracy.

Contribution

It introduces a topological perspective to thermometric precision, demonstrating how connectivity affects measurement accuracy at different temperature regimes.

Findings

Low connectivity enhances low-temperature thermometry.

High connectivity improves high-temperature thermometric performance.

Position measurement approaches energy measurement in precision.

Abstract

Temperature fluctuations of a finite system follows the Landau bound $\delta T^2 = T^2/C(T)$ where $C(T)$ is the heat capacity of the system. In turn, the same bound sets a limit to the precision of temperature estimation when the system itself is used as a thermometer. In this paper, we employ graph theory and the concept of Fisher information to assess the role of topology on the thermometric performance of a given system. We find that low connectivity is a resource to build precise thermometers working at low temperatures, whereas highly connected systems are suitable for higher temperatures. Upon modelling the thermometer as a set of vertices for the quantum walk of an excitation, we compare the precision achievable by position measurement to the optimal one, which itself corresponds to energy measurement.