A cycle decomposition and entropy production for circulant quantum Markov semigroups

We propose a definition of cycle representation for quantum Markov semigroups (qms) and quantum entropy production rate (qepr) in terms of the $\rho$-adjoint. We introduce the class of circulant qms, which admit non-equilibrium steady states but exhibit symmetries that allow us to compute explicitly the qepr, gain a deeper insight into the notion of cycle decomposition and prove that quantum detailed balance holds if and only if the qepr equals zero.