Algebraic conditions for convergence of a quantum Markov semigroup to a steady state

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 Sigma(l)(L-l*L(l)x - 2L(l)*xL(l) + xL(l)*L-l) + i[H, x] and a faithful normal invariant state rho. In this note we give new algebraic conditions for proving that T converges towards a steady state, possibly different from rho. Indeed, we show that this happens whenever the commutator of {H, L-l, L-l*vertical bar l >= 1} (i.e. its fixed point algebra) coincides with the commutator of {L-l, L-l*, delta(H)(L-l), delta(H)(L-l*), ..., delta(n)(H)(L-l), delta(n)(H)(L-l*)vertical bar l >= 1} (where delta(H)(X) = [H,X]) for some n >= 1. As an application we discuss the convergence to the unique invariant state of a spin chain model.