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We address the structure of the Liouvillian superoperator for a broad classof bosonic and fermionic Markovian open systems interacting with stationaryenvironments. We show that the accurate application of the partial secularapproximation in the derivation of the Bloch-Redfield master equation naturallyinduces a symmetry on the superoperator level, which may greatly reduce thecomplexity of the master equation by decomposing the Liouvillian superoperatorinto independent blocks. Moreover, we prove that, if the steady state of thesystem is unique, one single block contains all the information about it, andthat this imposes a constraint on the possible steady-state coherences of theunique state, ruling out some of them. To provide some examples, we show howthe symmetry appears for two coupled spins interacting with separate baths, aswell as for two harmonic oscillators immersed in a common environment. In bothcases the standard derivation and solution of the master equation issimplified, as well as the search for the steady state. Theblock-diagonalization does not appear when a local master equation is chosen.

Symmetry and block structure of the Liouvillian superoperator in partial secular approximation