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

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 $\delta T^2 \gtrsim T^4 e^{\Delta/T}$.