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

Minimal quantum dynamical semiguoup on a von Neumann algebra

Debashish Goswami, Kalyan B. Sinha

Published Jan 1, 1999• 0 citations
OpenAlex

Abstract

Given a formal unbounded generator, the minimal quantum dynamical semigroup on a von Neumann algebra is constructed. A set of equivalent necessary and sufficient conditions for the conservativity of the minimal semigroup is given and in the case when it is not conservative, a distinguished family of conservative perturbations of the semigroup is studied. Finally, some of these results are applied to the classical Markov semigroup with arbitrary state space.

Related papers

MINIMAL QUANTUM DYNAMICAL SEMIGROUP ON A VON NEUMANN ALGEBRA

Debashish Goswami, Kalyan B. Sinha

Given a formal unbounded generator, the minimal quantum dynamical semigroup on a von Neumann algebra is constructed. A set of equivalent necessary and sufficient conditions for the conservativity of the minimal semigroup is given and in the case when it is not conservative, a distinguished family of conservative perturbations of the semigroup is studied. Finally, some of these results are applied to the classical Markov semigroup with arbitrary state space.

Infinite Dimensional Analysis Quantum Probability and Related Topics•Jun 1, 1999•2 citations

Non-Conservative Minimal Quantum Dynamical Semigroups

Roberto B. Quezada

We give necessary and sufficient conditions for non-conservativity of a class of minimal quantum dynamical semigroups (qds). We extend some well known criteria for conservativity of minimal qds and show interesting relations of the conservativity problem with the von Neumann theory of the defect indices of a symmetric closed operator.

arXiv:math-ph/0112036•Dec 17, 2001•0 citations

QUANTUM DYNAMICAL SEMIGROUPS GENERATED BY NONCOMMUTATIVE UNBOUNDED ELLIPTIC OPERATORS

CHANGSOO BAHN, Chul Ki Ko, Yong‐Moon Park

We study quantum dynamical semigroups generated by noncommutative unbounded elliptic operators which can be written as Lindblad-type unbounded generators. Under appropriate conditions, we first construct the minimal quantum dynamical semigroups for the generators and then use Chebotarev and Fagnola's sufficient conditions for conservativity [1] to show that the semigroups are conservative. We then apply our results to a quantum mechanical system.

arXiv:math-ph/0505026•Reviews in Mathematical Physics•Jul 1, 2006•1 citation

Quantum dynamical semigroups

Kalyan B. Sinha, Debashish Goswami

Let us now restrict ourselves to the case when the general locally convex space X is replaced by a C * or a von Neumann algebra A , and study the implications of the complete positivity of a semigroup T t acting on it. Definition 3.0.1 A quantum dynamical semigroup (Q.D.S) on a C * -algebra A is a contractive semigroup T t of class C 0 such that each T t is a completely positive map from A to itself. T t is said to be conservative if T t (1) = 1 for all t ≥ 0. Generators of uniformly continuous quantum dynamical semigroups: the theorems of Lindblad and Christensen–Evans For a uniformly continuous semigroup on a von Neumann algebra A ⊆ B ( h ), we have the following result. Lemma 3.1.1 Let T t = e tL be a uniformly continuous contractive semigroup acting on A with L as the generator. Then T t is normal for each t if and only if L is ultra-strongly (and hence ultra-weakly) continuous on any norm-bounded subset of A . Proof : Let us first note that L is norm-bounded. If L is ultra-strongly continuous on bounded sets, then clearly e tL is ultra-strongly continuous on bounded sets for each t , and hence normal. For the converse, first note that for any t ≥ 0 and x ∈ A , we have Hence it is not difficult to see that Now suppose that x α is a net of elements in A such that x α strongly converges to x ∈ A and there exists positive constant M such that ∥ x α ∥ ≤ M for all α. Fix u ∈ h and ∈ > 0. Choose t 0 small enough so that ∥ L ∥ 2 M ∥ u ∥ t 0 ≤ ∈.

Cambridge University Press eBooks•Jan 25, 2007•0 citations

On The Generators of Quantum Dynamical Semigroups

Alexander Wiedemann

In recent years, digraph induced generators of quantum dynamical semigroups have been introduced and studied, particularly in the context of unique relaxation and invariance. We define the class of pair block diagonal generators, which allows for additional interaction coefficients but preserves the main structural properties. Namely, when the basis of the underlying Hilbert space is given by the eigenbasis of the Hamiltonian (for example the generic semigroups), then the action of the semigroup leaves invariant the diagonal and off-diagonal matrix spaces. In this case, we explicitly compute all invariant states of the semigroup. In order to define this class we provide a characterization of when the Gorini- Kossakowski-Sudarshan-Lindblad (GKSL) equation defines a proper generator when arbitrary Lindblad operators are allowed (in particular, they do not need to be trace- less as demanded by the GKSL Theorem). Moreover, we consider the converse con- struction to show that every generator naturally gives rise to a digraph, and that under certain assumptions the properties of this digraph can be exploited to gain knowledge of both the number and the structure of the invariant states of the corre- sponding semigroup. We also consider more general constructions on the von Neumann algebra of all bounded linear operators on a Hilbert space, perhaps infinite dimensional. In partic- ular, we prove that for every semigroup of Schwarz maps on such an algebra which has a subinvariant faithful normal state there exists an associated semigroup of con- tractions on the space of Hilbert-Schmidt operators of the Hilbert space. Moreover, we show that if the original semigroup is weak∗ continuous then the associated semigroup is strongly continuous. We introduce the notion of the extended generator of a semigroup on the bounded operators of a Hilbert space with respect to an or- thonormal basis of the Hilbert space. We describe this form of the generator of a quantum Markov semigroup on the von Neumann algebra of all bounded linear op- erators on a Hilbert space which has an invariant faithful normal state under the assumption that the generator of the associated semigroup has compact resolvent, or under the assumption that the minimal unitary dilation of the associated semigroup of contractions is compact.

•Jan 1, 2019•0 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
Given a formal unbounded generator, the minimal quantum dynamical semigroup on a von Neumann algebra is constructed. A set of equivalent necessary and sufficient conditions for the conservativity of the minimal semigroup is given and in the case when it is not conservative, a distinguished family of conservative perturbations of the semigroup is studied. Finally, some of these results are applied to the classical Markov semigroup with arbitrary state space.
Let us now restrict ourselves to the case when the general locally convex space X is replaced by a C * or a von Neumann algebra A , and study the implications of the complete positivity of a semigroup T t acting on it. Definition 3.0.1 A quantum dynamical semigroup (Q.D.S) on a C * -algebra A is a contractive semigroup T t of class C 0 such that each T t is a completely positive map from A to itself. T t is said to be conservative if T t (1) = 1 for all t ≥ 0. Generators of uniformly continuous quantum dynamical semigroups: the theorems of Lindblad and Christensen–Evans For a uniformly continuous semigroup on a von Neumann algebra A ⊆ B ( h ), we have the following result. Lemma 3.1.1 Let T t = e tL be a uniformly continuous contractive semigroup acting on A with L as the generator. Then T t is normal for each t if and only if L is ultra-strongly (and hence ultra-weakly) continuous on any norm-bounded subset of A . Proof : Let us first note that L is norm-bounded. If L is ultra-strongly continuous on bounded sets, then clearly e tL is ultra-strongly continuous on bounded sets for each t , and hence normal. For the converse, first note that for any t ≥ 0 and x ∈ A , we have Hence it is not difficult to see that Now suppose that x α is a net of elements in A such that x α strongly converges to x ∈ A and there exists positive constant M such that ∥ x α ∥ ≤ M for all α. Fix u ∈ h and ∈ > 0. Choose t 0 small enough so that ∥ L ∥ 2 M ∥ u ∥ t 0 ≤ ∈.
In recent years, digraph induced generators of quantum dynamical semigroups have been introduced and studied, particularly in the context of unique relaxation and invariance. We define the class of pair block diagonal generators, which allows for additional interaction coefficients but preserves the main structural properties. Namely, when the basis of the underlying Hilbert space is given by the eigenbasis of the Hamiltonian (for example the generic semigroups), then the action of the semigroup leaves invariant the diagonal and off-diagonal matrix spaces. In this case, we explicitly compute all invariant states of the semigroup. In order to define this class we provide a characterization of when the Gorini- Kossakowski-Sudarshan-Lindblad (GKSL) equation defines a proper generator when arbitrary Lindblad operators are allowed (in particular, they do not need to be trace- less as demanded by the GKSL Theorem). Moreover, we consider the converse con- struction to show that every generator naturally gives rise to a digraph, and that under certain assumptions the properties of this digraph can be exploited to gain knowledge of both the number and the structure of the invariant states of the corre- sponding semigroup. We also consider more general constructions on the von Neumann algebra of all bounded linear operators on a Hilbert space, perhaps infinite dimensional. In partic- ular, we prove that for every semigroup of Schwarz maps on such an algebra which has a subinvariant faithful normal state there exists an associated semigroup of con- tractions on the space of Hilbert-Schmidt operators of the Hilbert space. Moreover, we show that if the original semigroup is weak∗ continuous then the associated semigroup is strongly continuous. We introduce the notion of the extended generator of a semigroup on the bounded operators of a Hilbert space with respect to an or- thonormal basis of the Hilbert space. We describe this form of the generator of a quantum Markov semigroup on the von Neumann algebra of all bounded linear op- erators on a Hilbert space which has an invariant faithful normal state under the assumption that the generator of the associated semigroup has compact resolvent, or under the assumption that the minimal unitary dilation of the associated semigroup of contractions is compact.
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.