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Infinite Dimensional Analysis Quantum Probability and Related Topics

Quantum dynamical semigroups induced by Gaussian generalized Mehler semigroups

Un Cig Ji

Published Dec 20, 2025• 0 citations
Publisher

Abstract

Based on white noise theory, we first discuss the Gaussian generalized Mehler semigroups (on white noise functionals), along with their fundamental properties: invariant measures, invariant white noise operators, and mean ergodicity. Then, utilizing canonical topological isomorphisms between the spaces of two-variable white noise functionals and the spaces of white noise operators, we formulate a quantum analogue of the Gaussian generalized Mehler semigroups and discuss their fundamental properties. By establishing the notion of the positivity of white noise operators, we introduce the concept of quantum dynamical (Markov) semigroups acting on the space of white noise operators. Finally, we discuss the Gaussianity of the quantum dynamical semigroups induced by the Gaussian generalized Mehler semigroups. The construction of the quantum dynamical semigroups leads to the general notion of quantum dynamical semigroups on generalized operators.

DOI
10.1142/s0219025725500134

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Based on white noise theory, we first discuss the Gaussian generalized Mehler semigroups (on white noise functionals), along with their fundamental properties: invariant measures, invariant white noise operators, and mean ergodicity. Then, utilizing canonical topological isomorphisms between the spaces of two-variable white noise functionals and the spaces of white noise operators, we formulate a quantum analogue of the Gaussian generalized Mehler semigroups and discuss their fundamental properties. By establishing the notion of the positivity of white noise operators, we introduce the concept of quantum dynamical (Markov) semigroups acting on the space of white noise operators. Finally, we discuss the Gaussianity of the quantum dynamical semigroups induced by the Gaussian generalized Mehler semigroups. The construction of the quantum dynamical semigroups leads to the general notion of quantum dynamical semigroups on generalized operators.
Main purpose of this paper is to formulate canonical quantum continuity equations for white noise operators and to find their explicit unique solutions. By applying the continuity equations for white noise functionals and canonical topological isomorphisms between the spaces of white noise operators and the spaces of two-variable white noise functionals, we formulate canonical quantum continuity equations for white noise operators, and then we induce time dependent quantum evolution systems which are equivalent to the quantum continuity equations. Then we find explicit forms, in terms of the quantum analogues of generalized Fourier-Mehler transforms, of the unique solutions of the quantum continuity equations by solving the quantum evolution systems for white noise operators.
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.
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.
An intriguing and deep analogy between classical and quantum states is revealed using the notion of positive definite function. By Bochner's theorem, (classical) positive definite functions on phase space can be obtained by taking the Fourier transform of probability measures; similarly, quantum positive definite functions are the image via the Fourier-Plancherel operator of Wigner quasi-probability distributions. Considering the basic properties of positive definite functions — classical and quantum — one is led to define a class of semigroups of operators, the so-called classical-quantum semigroups. It is then natural to wonder whether they have any physical meaning. It turns out that the classical-quantum semigroups can also be obtained by dequantizing a certain class of quantum dynamical semigroups, namely, the classical noise semigroups. This correspondence fits in a more general group-theoretical framework in which a larger class of quantum dynamical semigroups, the twirling semigroups, can be suitably dequantized. Connections with quantum information science will be briefly discussed.