On two quantum versions of the detailed balance condition.

Quantum detailed balance conditions are often formulated as relationships between the generator of a quantum Markov semigroup and the generator of a dual semigroup with respect to a certain scalar product defined by an invariant state. In this paper we survey some results describing the structure of norm continuous quantum Markov semigroups on B(h) satisfying a quantum detailed balance condition when the duality is defined by means of pre-scalar products on B(h) of the form tr(\rho^{1-s}x*\rho^sy) (s\in [0, 1]) in order to compare the resulting quantum versions of the classical detailed balance condition. Moreover, we discuss the structure of generators of a quantum Markov semigroup which commute with the modular automorphism because this condition appears when we consider pre-scalar products with s\not= 1/2.