Thermodynamics of precision in quantum nonequilibrium steady states

Autonomous engines operating at the nanoscale can be prone to deleterious fluctuations in the heat and particle currents. Thermodynamic uncertainty relations (TURs) express a fundamental lower bound which translates a trade-off relation between precision and entropy production. Importantly, recent studies have shown that they can be violated in the quantum regime, thus motivating the search for analogous quantum counterparts. In this paper, we show that the geometry of quantum nonequilibrium steady states alone directly implies the existence of TUR, but with a looser bound, which is not violated by the above recent findings. The geometrical nature of this result makes it extremely general, establishing a fundamental limit for the thermodynamics of precision. Our proof is based on the McLennan-Zubarev ensemble, which provides an exact description of nonequilibrium steady states. We first prove that the entropy production of this ensemble can be expressed as a quantum relative entropy. The TURs are then shown to be a direct consequence of the Cramer-Rao bound, a fundamental result from parameter estimation theory. By combining techniques from many-body physics and information sciences, our approach also helps to shed light on the delicate relationship between quantum effects and current fluctuations in autonomous machines, where new general bound on the power output are found and discussed.