Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Infinite Dimensional Analysis Quantum Probability and Related Topics

A note on invariant states of Gaussian quantum Markov semigroups

Franco Fagnola, Damiano Poletti

Published May 17, 2024• 5 citations
Publisher

Abstract

Gaussian quantum Markov semigroup on the algebra of bounded operator on a [Formula: see text]-mode Fock space is characterized either by the property of their predual that preserves quantum Gaussian states. In this paper, show that gaussian QMSs can have only Gaussian invariant states if the spectrum of the drift matrix does not contain points on the imaginary axis. Moreover, in this case, any initial state converges in trace norm toward a unique gaussian invariant state.

DOI
10.1142/s0219025724400046

Related papers

Characterization of Gaussian quantum Markov semigroups

Damiano Poletti

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.

Infinite Dimensional Analysis Quantum Probability and Related Topics•Jul 12, 2022•10 citations

Gaussian quantum Markov semigroups on finitely many modes admitting a normal invariant state

Federico Girotti, Damiano Poletti

Gaussian quantum Markov semigroups (GQMSs) are of fundamental importance in modelling the evolution of several quantum systems. Moreover, they represent the noncommutative generalization of classical Orsntein-Uhlenbeck semigroups; analogously to the classical case, GQMSs are uniquely determined by a “drift” matrix Z and a “diffusion” matrix C , together with a displacement vector ζ . In this work, we completely characterize those GQMSs that admit a normal invariant state and we provide a description of the set of normal invariant states; as a side result, we are able to characterize quadratic Hamiltonians admitting a ground state in full generality and using a completely new approach compared to the ones used so far. Moreover, we study the behavior of such semigroups for long times: firstly, we clarify the relationship between the decoherence-free subalgebra and the spectrum of Z . Then, we prove that environment-induced decoherence takes place and that the dynamics approaches an Hamiltonian closed evolution for long times; we are also able to determine the speed at which this happens. Finally, we study convergence of ergodic means and recurrence and transience of the semigroup.

arXiv:2412.10020•Journal of Mathematical Analysis and Applications•Oct 17, 2025•1 citation

Second quantization and compactness of induced Gaussian quantum Markov semigroups

Zheng Li

We characterize compactness of the Hilbert-Schmidt semigroups induced by finite-mode Gaussian quantum Markov semigroups with a faithful normal invariant state. A unitary second-quantization representation for every $ s $-embedding shows that compactness at one positive time, compact resolvent, drift stability, and uniqueness of the normal invariant state are equivalent. These conditions imply immediate trace-class membership for all embeddings. We derive exact trace-norm and Hilbert-Schmidt norm formulas, an embedding-independent trace identity, and the generator spectrum with algebraic multiplicities. Examples show that compactness does not require the existence of a spectral gap and that a spectral gap need not imply compactness when the invariant state is not unique.

math.FACross-listedarXiv:2610.13358•Oct 12, 2026•0 citations

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

Julián Agredo, Franco Fagnola, Damiano Poletti

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.

Open Systems & Information Dynamics•Mar 1, 2021•19 citations

Symmetric Gaussian quantum Markov semigroups

Damiano Poletti, Emanuela Sasso, Veronica Umanità

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.

Journal of Mathematical Physics•Feb 1, 2026•0 citations

Irreducibility of Quantum Markov Semigroups, uniqueness of invariant states and related properties

Franco Fagnola, Federico Girotti

We present different characterizations of the notion of irreducibility for Quantum Markov Semigroups (QMSs) and investigate its relationship with other relevant features of the dynamics, such as primitivity, positivity improvement and relaxation; in particular, we show that irreducibility, primitivity and relaxation towards a faithful invariant density are equivalent when the semigroup admits an invariant density. Moreover, in the case of uniformly continuous QMSs, we present several useful ways of checking irreducibility in terms of the operators appearing in the generator in GKLS form. Our exposition is as much self-contained as possible, we present some well known results with elementary proofs (collecting all the relevant literature) and we derive new ones. We study both finite and infinite dimensional evolutions and we remark that many results only require the QMS to be made of Schwarz maps.

quant-pharXiv:2512.11517•Dec 15, 2025•0 citations
Gaussian quantum Markov semigroup on the algebra of bounded operator on a [Formula: see text]-mode Fock space is characterized either by the property of their predual that preserves quantum Gaussian states. In this paper, show that gaussian QMSs can have only Gaussian invariant states if the spectrum of the drift matrix does not contain points on the imaginary axis. Moreover, in this case, any initial state converges in trace norm toward a unique gaussian invariant state.
Gaussian quantum Markov semigroups (GQMSs) are of fundamental importance in modelling the evolution of several quantum systems. Moreover, they represent the noncommutative generalization of classical Orsntein-Uhlenbeck semigroups; analogously to the classical case, GQMSs are uniquely determined by a “drift” matrix Z and a “diffusion” matrix C , together with a displacement vector ζ . In this work, we completely characterize those GQMSs that admit a normal invariant state and we provide a description of the set of normal invariant states; as a side result, we are able to characterize quadratic Hamiltonians admitting a ground state in full generality and using a completely new approach compared to the ones used so far. Moreover, we study the behavior of such semigroups for long times: firstly, we clarify the relationship between the decoherence-free subalgebra and the spectrum of Z . Then, we prove that environment-induced decoherence takes place and that the dynamics approaches an Hamiltonian closed evolution for long times; we are also able to determine the speed at which this happens. Finally, we study convergence of ergodic means and recurrence and transience of the semigroup.
We characterize compactness of the Hilbert-Schmidt semigroups induced by finite-mode Gaussian quantum Markov semigroups with a faithful normal invariant state. A unitary second-quantization representation for every $ s $-embedding shows that compactness at one positive time, compact resolvent, drift stability, and uniqueness of the normal invariant state are equivalent. These conditions imply immediate trace-class membership for all embeddings. We derive exact trace-norm and Hilbert-Schmidt norm formulas, an embedding-independent trace identity, and the generator spectrum with algebraic multiplicities. Examples show that compactness does not require the existence of a spectral gap and that a spectral gap need not imply compactness when the invariant state is not unique.
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.
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 present different characterizations of the notion of irreducibility for Quantum Markov Semigroups (QMSs) and investigate its relationship with other relevant features of the dynamics, such as primitivity, positivity improvement and relaxation; in particular, we show that irreducibility, primitivity and relaxation towards a faithful invariant density are equivalent when the semigroup admits an invariant density. Moreover, in the case of uniformly continuous QMSs, we present several useful ways of checking irreducibility in terms of the operators appearing in the generator in GKLS form. Our exposition is as much self-contained as possible, we present some well known results with elementary proofs (collecting all the relevant literature) and we derive new ones. We study both finite and infinite dimensional evolutions and we remark that many results only require the QMS to be made of Schwarz maps.