Generators of Detailed Balance Quantum Markov Semigroups

For a quantum Markov semigroup T on the algebra B(h) with a faithful invariant state ρ, we can define an adjoint T̃ with respect to the scalar product determined by ρ. In this paper, we solve the open problems of characterising adjoints T̃ 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 L(x) = i[H,x] − 1 2 ∑ k(L ∗ kLkx − 2LkxLk + xLkLk) of the generator of T . We study the adjoint semigroup with respect to both scalar products 〈a, b〉 = tr(ρab) and 〈a, b〉 = tr(ρaρb). Kerwords: Quantum detailed balance, quantum Markov semigroup, Lindablad representation AMS Subject Classification: 46L55, 47D05, 82B10, 82C10, 81S25