GENERIC QUANTUM MARKOV SEMIGROUPS, CYCLE DECOMPOSITION AND DEVIATION FROM EQUILIBRIUM

Abstract

A generic quantum Markov semigroup [Formula: see text] of a d-level quantum open system with a faithful normal invariant state ρ admits a dual semigroup [Formula: see text] with respect to the scalar product induced by ρ. We show that the difference of the generators [Formula: see text] can be written as the sum of a derivation 2i[H, ⋅] and a weighted difference of automorphisms [Formula: see text] where [Formula: see text] is a family of cycles on the d levels of the system, wc are positive weights and [Formula: see text] 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.