On irreducibility of Gaussian quantum Markov semigroups

Abstract

The generator of a Gaussian quantum Markov semigroup on the algebra of bounded operator on a [Formula: see text]-mode Fock space is represented in a generalized GKLS form with an operator [Formula: see text] quadratic in creation and annihilation operators and Kraus operators [Formula: see text] linear in creation and annihilation operators. Kraus operators, commutators [Formula: see text] and iterated commutators [Formula: see text] up to the order [Formula: see text], as linear combinations of creation and annihilation operators determine a vector in [Formula: see text]. We show that a Gaussian quantum Markov semigroup is irreducible if such vectors generate [Formula: see text], under the technical condition that the domains of [Formula: see text] and the number operator coincide. Conversely, we show that this condition is also necessary if the linear space generated by Kraus operators and their iterated commutator with [Formula: see text] is fully non-commutative.