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 $\mathcal{L}$ are sums of other generators $\mathcal{L}_ω$ associated with positive Bohr frequencies $ω$ 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 $\mathcal{L}_ω$ so do with respect to the same invariant state.

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