# On the generators of quantum dynamical semigroups

@article{Lindblad1976OnTG, title={On the generators of quantum dynamical semigroups}, author={G{\"o}ran Lindblad}, journal={Communications in Mathematical Physics}, year={1976}, volume={48}, pages={119-130} }

The notion of a quantum dynamical semigroup is defined using the concept of a completely positive map. An explicit form of a bounded generator of such a semigroup onB(ℋ) is derived. This is a quantum analogue of the Lévy-Khinchin formula. As a result the general form of a large class of Markovian quantum-mechanical master equations is obtained.

#### 4,499 Citations

Uniqueness of the standard form of the generator of a quantum dynamical semigroup

- Mathematics
- 1980

Abstract All quantum dynamical semigroups so far constructed have generators of a standard form given below, but the uniqueness of this form has not been thoroughly investigated. We prove this under… Expand

A resolution of quantum dynamical semigroups

- Mathematics
- 2005

We consider a class of quantum dissipative systems governed by a one parameter completely positive maps on a von-Neumann algebra. We introduce a notion of recurrent and metastable projections for the… Expand

On the existence of stationary states for quantum dynamical semigroups

- Mathematics
- 2001

We provide two criteria on the existence of stationary states for quantum dynamical semigroups. The first one is based on the semigroup itself, while the second criterion is based on the generator… Expand

THE APPROACH TO EQUILIBRIUM OF A CLASS OF QUANTUM DYNAMICAL SEMIGROUPS

- Mathematics
- 1998

This paper deals with the asymptotic behavior of a quantum dynamical semigroup acting on the algebra of all linear bounded operators on a given Hilbert space. In practice, all these semigroups have a… Expand

Quantum stochastic semigroups and their generators

- Mathematics
- 1998

A rigged space characterisation of the unbounded generators of quantum completely positive (CP) stochastic semigroups is given. The general form and the dilation of the stochastic completely… Expand

Covariant quantum dynamical semigroups: Unbounded generators

- Mathematics
- 1998

A survey of probabilistic approaches to quantum dynamical semigroups with unbounded generators is given. An emphasis is made upon recent advances in the structural theory of covariant Markovian… Expand

Quantum dynamical semigroups and the neutron diffusion equation

- Mathematics
- 1977

Abstract We construct a class of quantum dynamical semigroups with unbounded generators by first obtaining the quantum mechanical analogue of the Feller minimal solution of the Fokker-Planck equation… Expand

On dilating quantum dynamical semigroups with classical brownian motion

- Mathematics
- 1986

We show that any quantum dynamical semigroup can be written with the help of the solution of an operator-valued classical stochastic differential equation. Moreover, this equation leads to a… Expand

Dilations of quantum dynamical semigroups with classical Brownian motion

- Mathematics
- 1987

We show that any quantum dynamical semigroup can be written with the help of the solution of a vector-valued classical stochastic differential equation. Moreover this equation leads to a natural… Expand

Structure of the Algebra of Effective Observables in Quantum Mechanics

- Physics
- 2000

Abstract A subclass of dynamical semigroups induced by the interaction of a quantum system with an environment is introduced. Such semigroups lead to the selection of a stable subalgebra of effective… Expand

#### References

SHOWING 1-10 OF 49 REFERENCES

On quantum statistical mechanics of non-Hamiltonian systems

- Mathematics
- 1972

Abstract An axiomatic definition of time evolution (dynamical semi-group) of a physical system has been given. A dynamical semi-group is defined as a one-parametersemi-group of linear endomorphisms… Expand

Markovian master equations

- Mathematics
- 1974

We give a rigorous proof that under certain technical conditions the memory effects in a quantum-mechanical master equation become negligible in the weak coupling limit. This is sufficient to show… Expand

General state changes in quantum theory

- Physics
- 1971

Abstract General state changes of quantum systems (operations) due to external interventions (measurements) are studied. The formalism developed in previous papers is simplified and generalized to… Expand

Harmonic Analysis of Operators on Hilbert Space

- Mathematics
- 1970

Contractions and Their Dilations.- Geometrical and Spectral Properties of Dilations.- Functional Calculus.- Extended Functional Calculus.- Operator-Valued Analytic Functions.- Functional Models.-… Expand

On the connection of nonequilibrium information thermodynamics with non-hamiltonian quantum mechanics of open systems

- Physics
- 1975

Abstract A critical and improved version of the non-Hamiltonian quantum mechanics and nonequilibrium information thermodynamics is presented. It has been shown that the latter is connected with the… Expand

Quantum stochastic processes

- Physics
- 1969

In order to describe rigorously certain measurement procedures, where observations of the arrival of quanta at a counter are made throughout an interval of time, it is necessary to introduce the… Expand

Difficulty with a kinematic concept of unstable particles: the SZ.-Nagy extension and the Matthews-Salam-Zwanziger representation

- Physics
- 1971

We discuss the possibility of describing unstable systems, or dissipative systems in general, by vectors in a Hilbert space, evolving in time according to some non-unitary group or semigroup of… Expand

The harmonic oscillator in a heat bath

- Physics
- 1973

We study the time evolution of a quantum-mechanical harmonic oscillator in interaction with an infinite heat bath, which is supposed to be initially in the canonical equilibrium at some temperature.… Expand

On the identity of three generalized master equations

- Mathematics
- 1964

Abstract Three apparently different quantum mechanical master equations, derived by Prigogine and Resibois, by Montroll, and independently by Nakajima and Zwanzig, are shown to be identical. The… Expand

The Bloch equations

- Mathematics
- 1974

We consider a spinor interacting with a heat bath of harmonic oscillators in equilibrium and we prove that the phenomenological Bloch equations for time development are satisfied exactly if the spin… Expand