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Open Systems & Information Dynamics

Gaussian Quantum Markov Semigroups on a One-Mode Fock Space: Irreducibility and Normal Invariant States

Julián Agredo, Franco Fagnola, Damiano Poletti

Published Mar 1, 2021• 19 citations
Publisher

Abstract

We consider the most general Gaussian quantum Markov semigroup on a one-mode Fock space, discuss its construction from the generalized GKSL representation of the generator. We prove the known explicit formula on Weyl operators, characterize irreducibility and its equivalence to a Hörmander type condition on commutators and establish necessary and sufficient conditions for existence and uniqueness of normal invariant states. We illustrate these results by applications to the open quantum oscillator and the quantum Fokker-Planck model.

DOI
10.1142/s1230161221500013

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We consider the most general Gaussian quantum Markov semigroup on a one-mode Fock space, discuss its construction from the generalized GKSL representation of the generator. We prove the known explicit formula on Weyl operators, characterize irreducibility and its equivalence to a Hörmander type condition on commutators and establish necessary and sufficient conditions for existence and uniqueness of normal invariant states. We illustrate these results by applications to the open quantum oscillator and the quantum Fokker-Planck model.
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.
We investigate symmetric Gaussian Quantum Markov Semigroups (G-QMSs), a class of quantum dynamical semigroups acting on the algebra of bounded operators over a bosonic Fock space, whose generators are given in generalized GKLS form with linear and quadratic terms in canonical operators. Our main goal is to characterize the symmetry of such semigroups with respect to a faithful invariant Gaussian state, in terms of explicit algebraic conditions on the generator’s parameters. We provide necessary and sufficient conditions for GNS-symmetry using the action of the semigroup on Weyl operators, and show that under natural assumptions, every symmetric G-QMS is unitarily equivalent to a direct sum of Ornstein–Uhlenbeck-type semigroups. The results are further linked to modular theory by confirming the well-known fact that GNS-symmetry implies commutation with the modular automorphism group. Technical appendices discuss the role of Bogoliubov and metaplectic transformations in simplifying the structure of symmetric semigroups.
We study regularisation and irreducibility properties of Gaussian quantum Markov semigroups (GQMSs) acting on continuous-variable quantum systems. We first identify a natural notion of regularity for operators in this setting, which allows us to formulate and characterise the smoothing effects of GQMSs in terms of algebraic conditions involving the drift and quantum diffusion matrices. These conditions establish a connection with the controllability theory of quantum linear systems and with the structure of decoherence-free subsystems. We then characterise irreducibility through several equivalent algebraic criteria: one formulated in terms of the drift and quantum diffusion matrices, one in terms of the operators appearing in the generalised GKLS representation of the generator, and a third given by a quantum analogue of Hörmander's condition. A central and somewhat surprising consequence is that, in contrast with the classical case, irreducibility is strictly stronger than conditions ensuring regularisation. Our results provide an algebraic framework for analysing these properties and lay the groundwork for a broader study of reducible Gaussian quantum Markov semigroups and, more generally, more general relevant quantum Markov semigroups on continuous-variable systems.
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