  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.
