Quantum detailed balance conditions with time reversal: three-level system

A generic quantum Markov semigroup $\T$ of a $d$-level quantum open system with a faithful normal invariant state $\rho$ admits a dual semigroup $\tT$ with respect to the scalar product induced by $\rho$. We show that the difference of the generators $\Ll-\widetilde{\Ll}$ can be written as the sum of a derivation $2i[H,\cdot]$ and a weighted difference of automorphisms \[ \sum_{c\in\C}w_c\rho^{-1/2}\left(\frac{1}{d}\sum_{j=1}^{d} \left(U^{(c)*}_j xU^{(c)}_j-U^{(c)}_jx U^{(c)*}_j\right)\right)\rho^{-1/2} \] where $\C$ is a family of cycles on the $d$ levels of the system, $w_c$ are positive weights and $U^{(c)}_j$ are unitaries. This formula allows us to represent the deviation from equilibrium (in a ``small'' time interval) as the superposition of cycles of the system where the difference between the forward and backward evolution is written as the difference of a reversible evolution and its time reversal. Moreover, it generalises cycle decomposition of Markov jump processes. We also find a similar formula with partial isometries instead of unitaries.