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

Lecture notes in pure and applied mathematics

Some Results on Invariant States for Quantum Markov Semigroups

Hisao Watanabe, Franco Fagnola

Published Apr 5, 2002• 0 citations
Publisher
DOI
10.1201/9780203910177.ch12

Related papers

Notes on the Qualitative Behaviour of Quantum Markov Semigroups

Franco Fagnola, Rolando Rebolledo

Lecture notes in mathematics•Jan 1, 2006•31 citations

Invariant Projections for Covariant Quantum Markov Semigroups

Franco Fagnola, Emanuela Sasso, Veronica Umanità

In this paper we investigate consequences of covariance of a uniformly Quantum Markov Semigroup, under a group action, on the structure of its minimal invariant projections.We obtain that, under suitable hypotheses, minimal invariant projections correspond to irreducible sub-representations in which the initial covariant representation is decomposed.We apply this results in the study circulant Quantum Markov Semigroups.

Journal of Stochastic Analysis•Nov 6, 2020•1 citation

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

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

Some Markov semigroups in quantum probability

Alberto Barchielli

Lecture notes in mathematics•Jan 1, 1990•3 citations

Quantum Markov Semigroups and Quantum Markov Flows

Franco Fagnola

•Jan 1, 1999•25 citations
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 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.