THE DECOHERENCE-FREE SUBALGEBRA OF A QUANTUM MARKOV SEMIGROUP WITH UNBOUNDED GENERATOR

Abstract

Let T be a quantum Markov semigroup on B(h) with a faithful normal invariant state.. The decoherence-free subalgebra N(T) of T is the biggest subalgebra of B(h) where the completely positive maps T(t) act as homomorphisms. When T is the minimal semigroup whose generator is represented in a generalised GKSL form L(x) = -1/2 Sigma(l)(L(l)*L(l)x-2L(l)*xL(l)+xL(l)*L(l))+i[H, x], with possibly unbounded H, L(l), we show that N(T) coincides with the generalised commutator of {e(-itH) L(l)e(itH), e(-itH) L(l)*e(itH) vertical bar l >= 1, t >= 0} under some natural regularity conditions.

As a corollary we derive simple sufficient algebraic conditions for convergence towards a steady state based on multiple commutators of H and L(l).

We give examples of quantum Markov semigroups B(h), with h infinite-dimensional, having a non-trivial decoherence-free subalgebra.