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lsdragon

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- 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?

> 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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