  Markovian master equations (formally known as quantum dynamical semigroups)
can be used to describe the evolution of a quantum state $\rho$ when in contact
with a memoryless thermal bath. This approach has had much success in
describing the dynamics of real-life open quantum systems in the lab. Such
dynamics increase the entropy of the state $\rho$ and the bath until both
systems reach thermal equilibrium, at which point entropy production stops. Our
main result is to show that the entropy production at time $t$ is bounded by
the relative entropy between the original state and the state at time $2t$. The
bound puts strong constraints on how quickly a state can thermalise, and we
prove that the factor of $2$ is tight. The proof makes use of a key physically
relevant property of these dynamical semigroups -- detailed balance, showing
that this property is intimately connected with the field of recovery maps from
quantum information theory. We envisage that the connections made here between
the two fields will have further applications. We also use this connection to
show that a similar relation can be derived when the fixed point is not
thermal.
