  In classical Markov jump processes, current fluctuations can only be reduced
at the cost of increased dissipation. To explore how quantum effects influence
this trade-off, we analyze the uncertainty of steady-state currents in
Markovian open quantum systems. We first consider three instructive examples
and then systematically minimize the product of uncertainty and entropy
production for small open quantum systems. As our main result, we find that the
thermodynamic cost of reducing fluctuations can be lowered below the classical
bound by coherence. We conjecture that this cost can be made arbitrarily small
in quantum systems with sufficiently many degrees of freedom. Our results
thereby provide a general guideline for the design of thermal machines in the
quantum regime that operate with high thermodynamic precision, meaning low
dissipation and small fluctuations around average values.
