  We give an explicit entropy production formula for a class of quantum Markov
semigroups, arising in the weak coupling limit of a system coupled with
reservoirs, whose generators $\mathcal{L}$ are sums of other generators
$\mathcal{L}_\omega$ associated with positive Bohr frequencies $\omega$ of the
system. As a consequence, we show that any such semigroup satisfies the quantum
detailed balance condition with respect to an invariant state if and only if
all semigroups generated by each $\mathcal{L}_\omega$ so do with respect to the
same invariant state.
