Achieving the Landau bound to precision of quantum thermometry in systems with vanishing gap

TL;DR

This paper demonstrates that quantum systems with vanishing energy gaps can achieve the fundamental Landau bound for temperature measurement precision, unlike gapped systems where precision deteriorates exponentially.

Contribution

It shows that the Landau bound can be reached in microscopic quantum systems with vanishing gaps, extending classical thermometry bounds to quantum regimes.

Findings

Vanishing gap systems attain the Landau bound for temperature precision.

Gapped systems' temperature estimation precision diverges exponentially with decreasing temperature.

Energy measurement suffices to reach optimal thermometric precision in gapless quantum systems.

Abstract

We address estimation of temperature for finite quantum systems at thermal equilibrium and show that the Landau bound to precision $\delta T^2 \propto T^2$, originally derived for a classical {\em not too small} system being a portion of a large isolated system at thermal equilibrium, may be also achieved by energy measurement in microscopic {\em quantum} systems exhibiting vanishing gap as a function of some control parameter. On the contrary, for any quantum system with a non-vanishing gap $\Delta$, precision of any temperature estimator diverges as $\delta T^2 \gtrsim T^4 e^{\Delta/T}$.