  In quantum thermodynamics, entropy production is usually defined in terms of
the quantum relative entropy between two states. We derive a lower bound for
the quantum entropy production in terms of the mean and variance of quantum
observables, which we will refer to as a thermodynamic uncertainty relation
(TUR) for the entropy production. In the absence of coherence between the
states, our result reproduces classic TURs in stochastic thermodynamics. For
the derivation of the TUR, we introduce a lower bound for a quantum
generalization of the $\chi^2$ divergence between two states and discuss its
implications for stochastic and quantum thermodynamics, as well as the limiting
case where it reproduces the quantum Cram\'er-Rao inequality.
