Generic Quantum Markov Semigroups, Cycle Decomposition and Deviation from Equilibrium

A generic quantum Markov semigroup T of a d-level quantum open system with a faithful normal invariant state $\rho$ admits a dual semigroup $\widetilde{T}$ with respect to the scalar product induced by $\rho$. We show that the difference of the generators $L-\widetilde{L}$ can be written as the sum of a derivation 2i[H, ]$ and a weighted difference of automorphisms $\sum_{c\in\C}w_c\rho^{-1/2}\left(\frac{1{d}\sum_{j=1}^{d} \left(U^{(c)*}_j xU^{(c)}_j-U^{(c)}_jxU^{(c)*}_j\right)\right)\rho^{-1/2}$ 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.