GENERATORS OF DETAILED BALANCE QUANTUM MARKOV SEMIGROUPS.

For a quantum Markov semigroup ${\mathcal T}$ on the algebra ${\mathcal B} ({\mathsf h})$ with a faithful invariant state ρ, we can define an adjoint $\widetilde T$ with respect to the scalar product determined by ρ. In this paper, we solve the open problems of characterizing adjoints $\widetilde T$ that are also a quantum Markov semigroup and satisfy the detailed balance condition in terms of the operators H, Lk in the Gorini–Kossakowski–Sudarshan–Lindblad representation ${\mathcal L} (x) =i [H, x] -\frac{1}{2} \sum_k (L^*_k L_k x -2L^*_k x L_k +xL^*_k L_k)$ of the generator of ${\mathcal T}$. We study the adjoint semigroup with respect to both scalar products 〈a, b〉 =tr (ρa*b) and 〈a, b〉 =tr (ρ1/2a*ρ1/2 b). [ABSTRACT FROM AUTHOR]