Energy-temperature uncertainty relation in quantum thermodynamics

TL;DR

This paper derives a generalized thermodynamic uncertainty relation for quantum and classical systems, revealing how quantum fluctuations and interactions influence temperature measurement precision at the nanoscale.

Contribution

It introduces a new uncertainty relation incorporating quantum effects and interactions, advancing understanding of temperature estimation in quantum thermodynamics.

Findings

Quantum fluctuations increase temperature uncertainty.

The uncertainty is constrained by heat capacity and dissipation.

Interactions and non-commutativity affect measurement precision.

Abstract

It is known that temperature estimates of macroscopic systems in equilibrium are most precise when their energy fluctuations are large. However, for nanoscale systems deviations from standard thermodynamics arise due to their interactions with the environment. Here we include such interactions and, using quantum estimation theory, derive a generalised thermodynamic uncertainty relation valid for classical and quantum systems at all coupling strengths. We show that the non-commutativity between the system's state and its effective energy operator gives rise to quantum fluctuations that increase the temperature uncertainty. Surprisingly, these additional fluctuations are described by the average Wigner-Yanase-Dyson skew information. We demonstrate that the temperature's signal-to-noise ratio is constrained by the heat capacity plus a dissipative term arising from the non-negligible interactions. These findings shed light on the interplay between classical and non-classical fluctuations in quantum thermodynamics and will inform the design of optimal nanoscale thermometers.