Entropy Production and Detailed Balance for a Class of Quantum Markov Semigroups

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 are sums of other generators L_i associated with positive Bohr frequencies 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 L_i do so with respect to the same invariant state