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We show that all Lindblad operators (i.e. generators of quantum semigroups)on a finite-dimensional Hilbert space satisfying the detailed balance conditionwith respect to the thermal equilibrium state can be written as a gradientsystem with respect to the relative entropy.  We discuss also thermodynamically consistent couplings to macroscopicsystems, either as damped Hamiltonian systems with constant temperature or asGENERIC systems. In particular we discuss the coupling of a quantum dot coupledto macroscopic charge carriers.

An entropic gradient structure for Lindblad equations and couplings of quantum systems to macroscopic models