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Infinite Dimensional Analysis Quantum Probability and Related Topics

Characterization of Gaussian quantum Markov semigroups

Damiano Poletti

Published Jul 12, 2022• 10 citations
Publisher

Abstract

We give a characterization of QMSs on the Bosonic Fock Space [Formula: see text] whose predual preserves the set of gaussian states. We show they can be obtained via certain generalized GKLS generators and they satisfy an explicit formula for their action on Weyl operators.
DOI
10.1142/s021902572250014x

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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.