Temperature uncertainty relation in non-equilibrium thermodynamics

Temperature uncertainty of a quantum system in canonical ensemble is inversely determined by its energy fluctuation, which is known as the temperature-energy uncertainty relation. No such uncertainty relation was discovered for a non-equilibrium open quantum system. In this article, we derive a universal temperature uncertainty relation for general non-equilibrium processes. We find that it is the fluctuation of heat, which is defined as the change in bath energy, determines the temperature uncertainty in non-equilibrium case. Specifically, the heat is divided into trajectory heat and backaction heat, which are associated with the system's trajectory of evolution and the backaction of partial measurement on system, respectively. Based on this decomposition, we reveal that both correlations between system and bath's process function and state function are the resources for enhancing temperature precision. Our findings are conductive to design ultrahigh sensitive quantum thermometer.