Abstract

Let T be a uniformly continuous quantum Markov semigroup on B(h) with generator represented in a standard GKSL form L(x) = -1/2 ∑l (L*l x - 2L*l xLl + xLρlLl) + i[H,x] and a faithful normal invariant state ρ. In this note we give new algebraic conditions for proving that T converges towards a steady state, possibly different from ρ. Indeed, we show that this happens whenever the commutator of {H, Ll, L*l|l ≥ 1} (i.e. its fixed point algebra) coincides with the commutator of {Ll, L*l, δH (L l), δH (L*l),..., δn H (Ll), δnH (L*l)|l ≥ 1} (where δH(X) = [H, X]) for some n < 1. As an application we discuss the convergence to the unique invariant state of a spin chain model. © 2008 World Scientific Publishing Company.