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

Birkhäuser Basel eBooks

Quantum Markov Semigroups and their Stationary States

Franco Fagnola, Rolando Rebolledo

Published Jan 1, 2003• 18 citations
Publisher

Abstract

The notion of a Quantum Markov Semigroup (QMS) is an extension of classical Markov semigroups motivated by the study of open quantum systems. Indeed, a number of physicists interested in open quantum systems introduced the notion of Quantum Dynamical Semigroups in the Seventies. Since then the subject has been extensively discussed and improved by both physicists and mathematicians, especially by the founders of Quantum Stochastic Analysis. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

DOI
10.1007/978-3-0348-8018-3_6

Related papers

Complete Positivity and Open Quantum Systems

Rolando Rebolledo

The interpretation provided by Schrödinger of quantum mechanics through classical stochastic processes is the root of Euclidean quantum mechanics founded on Bernstein processes. Open quantum systems have been extensively investigated during the last three decades and have motivated new probabilistic interpretations for quantum dynamics. This article, mainly addressed to classical probabilists, gives a short introduction to quantum dynamical semigroups and quantum Markov flows as a natural extension of Schrödinger ideas in noncommutative stochastic analysis. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Birkhäuser Boston eBooks•Jan 1, 2001•2 citations

Quantum Markov Semigroups

Mou-Hsiung Chang

In this chapter, we define and explore basic properties of a quantum Markov semigroup { I t , t ≥ 0} of linear maps on the C* -algebra or von Neumann algebra A that characterizes the quantum system. The quantum Markov semigroup (QMS) plays a key role in describing quantum Markov processes that are to be explored in the subsequent chapters. The concept of QMS extends the semigroup of probability transition operators { T t , t ≥ 0} for a classical Markov process to its noncommutative counterpart. Suppose the classical Markov process { X t , t ≥ 0} is defined on the complete filtered classical probability space (Ω, Ƒ, P ; {Ƒ t , t ≥ 0}) and with values in a measurable space (핏, B (X)). Recall that for each t ≥ 0, the probability transition operator T t : L ∞ (X, B (X)) → L ∞ (X, B (X)) by ( T t f )( X s ) = E x [ f ( X s + t ) | Ƒ s ] ∀ s , t ≥ 0. A semigroup of linear maps on the C* -algebra or von Neumann algebra A is said to be a quantum dynamical semigroup (QDS) if (i) I 0 = J (the identity operator on A ); (ii) I t I s = I t + s for all t , s ≥ 0; (iii) I t is completely positive for each t ≥ 0; and (iv) I t is σ weakly continuous on A , i.e., a ↦ tr ( ρI t ( a )) is continuous from A to ℂ for each ρ ∈ S ( A ) (the space of quantum states) and for each t ≥ 0. If in addition I t ( I ) = I , (respectively, I t ( I ) ≤ I ) for all t ≥ 0, then the QDS is said to be a quantum Makrov semigroup (QMS) (respectively, quantum sub-Markov semigroup).

Cambridge University Press eBooks•Feb 9, 2015•0 citations

Quasi-Stationary Normal States for Quantum Markov Semigroups

Ameur Dhahri, Franco Fagnola, Federico Girotti +1 more

We introduce a notion of quasi-stationary state (QSS) in the context of quantum Markov semigroups that generalizes the one of quasi-stationary distribution in the case of classical Markov chains. We provide an operational interpretation of QSSs using the theory of direct and indirect quantum measurements. Moreover, we prove that there is a connection between QSSs and spectral properties of the quantum Markov semigroup. Finally, we discuss some examples which, despite their simplicity, already show interesting features.

arXiv:2508.06396•Open Systems & Information Dynamics•Sep 1, 2025•0 citations

Quantum Stochastic Processes

Luigi Accardi

Let ℬ be *-algebra with identity (usually it wil be a C*- or a W*-algebra). A quantum stochastic process over ℬ indexed by ℝ is defined by a triple { A , (j t ) t∈ℝ , φ } where A is a *-algebra with identity. j t : ℬ ↪ A is an embedding (t∈ℝ). φ is a State on A . These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Birkhäuser Boston eBooks•Jan 1, 1985•226 citations

Unraveling open quantum systems: classical reductions and classical dilations of quantum markov semigroups

Rolando Rebolledo

A number of results connecting quantum and classical Markov semigroups, as well as their dilations is reported. The method presented here is based on the analysis of the structure of the semigroup generator. In particular, measure-valued processes appear as a combination of classical reduction and classical dilation of a given quantum Markov semigroup.

Confluentes Mathematici•May 22, 2009•6 citations

A Simple Singular Quantum Markov Semigroup

Franco Fagnola

We construct a quantum Markov semigroup on the von Neumann algebra B ( L 2 (ℝ + ;ℂ)) of all bounded operators on a Hilbert space L 2 (ℝ + ;ℂ) with the following property: the biggest *-subalgebra of B ( L 2 (ℝ + ;ℂ)) contained in the domain of the infinitesimal generator is not σ-weakly dense. Our semigroup is an extension of a classical Markov semigroup on L ∞ (ℝ + ;ℂ) with the same property. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Birkhäuser Boston eBooks•Jan 1, 2000•10 citations
The notion of a Quantum Markov Semigroup (QMS) is an extension of classical Markov semigroups motivated by the study of open quantum systems. Indeed, a number of physicists interested in open quantum systems introduced the notion of Quantum Dynamical Semigroups in the Seventies. Since then the subject has been extensively discussed and improved by both physicists and mathematicians, especially by the founders of Quantum Stochastic Analysis. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
The interpretation provided by Schrödinger of quantum mechanics through classical stochastic processes is the root of Euclidean quantum mechanics founded on Bernstein processes. Open quantum systems have been extensively investigated during the last three decades and have motivated new probabilistic interpretations for quantum dynamics. This article, mainly addressed to classical probabilists, gives a short introduction to quantum dynamical semigroups and quantum Markov flows as a natural extension of Schrödinger ideas in noncommutative stochastic analysis. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
In this chapter, we define and explore basic properties of a quantum Markov semigroup { I t , t ≥ 0} of linear maps on the C* -algebra or von Neumann algebra A that characterizes the quantum system. The quantum Markov semigroup (QMS) plays a key role in describing quantum Markov processes that are to be explored in the subsequent chapters. The concept of QMS extends the semigroup of probability transition operators { T t , t ≥ 0} for a classical Markov process to its noncommutative counterpart. Suppose the classical Markov process { X t , t ≥ 0} is defined on the complete filtered classical probability space (Ω, Ƒ, P ; {Ƒ t , t ≥ 0}) and with values in a measurable space (핏, B (X)). Recall that for each t ≥ 0, the probability transition operator T t : L ∞ (X, B (X)) → L ∞ (X, B (X)) by ( T t f )( X s ) = E x [ f ( X s + t ) | Ƒ s ] ∀ s , t ≥ 0. A semigroup of linear maps on the C* -algebra or von Neumann algebra A is said to be a quantum dynamical semigroup (QDS) if (i) I 0 = J (the identity operator on A ); (ii) I t I s = I t + s for all t , s ≥ 0; (iii) I t is completely positive for each t ≥ 0; and (iv) I t is σ weakly continuous on A , i.e., a ↦ tr ( ρI t ( a )) is continuous from A to ℂ for each ρ ∈ S ( A ) (the space of quantum states) and for each t ≥ 0. If in addition I t ( I ) = I , (respectively, I t ( I ) ≤ I ) for all t ≥ 0, then the QDS is said to be a quantum Makrov semigroup (QMS) (respectively, quantum sub-Markov semigroup).
We introduce a notion of quasi-stationary state (QSS) in the context of quantum Markov semigroups that generalizes the one of quasi-stationary distribution in the case of classical Markov chains. We provide an operational interpretation of QSSs using the theory of direct and indirect quantum measurements. Moreover, we prove that there is a connection between QSSs and spectral properties of the quantum Markov semigroup. Finally, we discuss some examples which, despite their simplicity, already show interesting features.
We construct a quantum Markov semigroup on the von Neumann algebra B ( L 2 (ℝ + ;ℂ)) of all bounded operators on a Hilbert space L 2 (ℝ + ;ℂ) with the following property: the biggest *-subalgebra of B ( L 2 (ℝ + ;ℂ)) contained in the domain of the infinitesimal generator is not σ-weakly dense. Our semigroup is an extension of a classical Markov semigroup on L ∞ (ℝ + ;ℂ) with the same property. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.