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For a quantum Markov semigroup T on the algebra B(h) with a faithful invariant state ρ, we can define an adjoint T with respect to the scalar product determined by ρ. In this paper, we solve the open problems of characterising adjoints T that are also a quantum Markov semigroup and satisfy the detailed balance condition in terms of the operators H,We study the adjoint semigroup with respect to both scalar products a, b = tr(ρa * b) and a, b = tr(ρ 1/2 a * ρ 1/2 b).

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This article introduces a concept of subharmonic projections for a quantum Markov semigroup, in view of characterizing the support projection of a stationary state in terms of the semigroup generator. These results, together with those of our previous article [J. Math. Phys. 42, 1296 (2001)], lead to a method for proving the existence of faithful stationary states. This is often crucial in the analysis of ergodic properties of quantum Markov semigroups. The method is illustrated by applications to physical models.

We find the structure of generators of norm continuous quantum Markov semigroups on B(h) that are symmetric with respect to the scalar product tr(ρ 1/2 x * ρ 1/2 y) induced by a faithful normal invariant state invariant state ρ and satisfy two quantum generalisations of the classical detailed balance condition related with this non-commutative notion of symmetry: the socalled standard detailed balance condition and the standard detailed balance condition with an antiunitary time reversal.

The structure of uniformly continuous quantum Markov semigroups with atomic decoherence-free subalgebra is established providing a natural decomposition of a Markovian open quantum system into its noiseless (decoherencefree) and irreducible (ergodic) components. This leads to a new characterisation of the structure of invariant states and a new method for finding decoherence-free subsystems and subspaces. Examples are presented to illustrate these results.

We provide two criteria on the existence of stationary states for quantum dynamical semigroups. The first one is based on the semigroup itself, while the second criterion is based on the generator which is in general unbounded and interpreted as a sesquilinear form. These results are illustrated by physical examples drawn from quantum optics.