  Autonomous engines operating at the nano-scale can be prone to deleterious
fluctuations in the heat and particle currents which increase, for fixed power
output, the more reversible the operation regime is. This fundamental trade-off
between current fluctuations and entropy production forms the basis of the
recently formulated thermodynamic uncertainty relations (TURs). However, these
relations have so far only been derived for classical Markovian systems and can
be violated in the quantum regime. In this paper we show that the geometry of
quantum non-equilibrium steady-states alone, already directly implies the
existence of a TUR, but with a looser bound. 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 non-equilibrium 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 quantum 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.
