Energy-temperature uncertainty relation in quantum thermodynamics

Abstract

Much like Heisenberg’s uncertainty principle in quantum mechanics, there exists a thermodynamic uncertainty relation in classical statistical mechanics that limits the simultaneous estimation of energy and temperature for a system in equilibrium. However, for nanoscale systems deviations from standard thermodynamics arise due to non-negligible interactions with the environment. Here we include 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 additional quantum fluctuations that increase the uncertainty in temperature and modify the heat capacity. Surprisingly, these quantum fluctuations are described by the average Wigner-Yanase-Dyson skew information, a quantity intimately connected to measures of coherence. For temperature estimation we demonstrate that the optimal signal-to-noise ratio is constrained not only by the heat capacity, but an additional dissipative term arising from the non-negligible interactions. Practically this will inform the design of optimal nanoscale thermometers. On the fundamental side the results shed light on the interplay between classical and non-classical fluctuations in quantum thermodynamics.