How to derive the quantum detailed balance condition?

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• lsdragon
In summary, the conversation discusses the detailed balance condition for non-Hamiltonian systems, specifically for Markov open quantum systems. The detailed balance condition is defined as the generator ##L## being a normal operator in the Hilbert space ##\mathcal{B}_{\rho_0}(\mathcal{H})##. This is derived from the classical detailed balance condition ##p_{ij}\pi_j = p_{ji}\pi_i##, and the author also presents a quantum analogy for detailed balance. However, there is a question on how to derive definition 2 from the classical version of detailed balance.
lsdragon
TL;DR Summary
I want some help to get the definition of quantum detailed balance condition from analogy of classical detailed balance condition
In the "On The detailed balance conditions for non-Hamiltonian systems", I learned that for a Markov open quantum system to satisfying the master equation with the Liouvillian superoperators, the detailed balance condition will be

> Definition 2: The open quantum Markovian system (##dim(\mathcal{H}) < \infty##) obeys the detailed balance principle if the generator ##L## in Heisenberg picture is a normal operator in Hilbert space ##\mathcal{B}_{\rho_0}(\mathcal{H})## (see Definition 1).

>Definition 1: ##\mathcal{B}_{\rho_0}(\mathcal{H})## denotes the Hilbert space of all linear operators on the finite-dimensional Hilbert space ##\mathcal{H}## with the scalar product defined by the formula
$$\langle A, B\rangle = Tr(A^\dagger B \rho_0), A,B \in \mathcal{B}_{\rho_0}(\mathcal{H})$$
where ##\rho_0## is a fixed state (density matrix) and ## \rho_0 > 0##.

The ##L## is the adjoint operator, defined with respect to definition 1, of the Liouvillian superoperator ##\mathcal{L}##, such that
$$\frac{d \rho}{d t} = \mathcal{L} \rho \\ \frac{d A}{d t} = L A, A\in \mathcal{B}_{\rho_0}(\mathcal{H}).$$

The author started from the classical detailed balance condition ##p_{ij}\pi_j = p_{ji}\pi_i## and finally get to definition 2.

For me, I will write the quantum analogy of detailed balance as
$$\langle A,L(B) \rangle = \langle B, L(A)\rangle .$$
I can not get the normality of ##L## from the above definition.
Then, my question is that how can we get to definition 2 starting from the classical version of detailed balance?

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