Quantum Markov semigroup


In quantum mechanics, a quantum Markov semigroup describes the dynamics in a Markovian open quantum system. The axiomatic definition of the prototype of quantum Markov semigroups was first introduced by A. M. Kossakowski in 1972, and then developed by V. Gorini, A. M. Kossakowski, E. C. G. Sudarshan and Göran Lindblad in 1976.

Motivation

An ideal quantum system is not realistic because it should be completely isolated while, in practice, it is influenced by the coupling to an environment, which typically has a large number of degrees of freedom. A complete microscopic description of the degrees of freedom of the environment is typically too complicated. Hence, one looks for simpler descriptions of the dynamics of the open system. In principle, one should investigate the unitary dynamics of the total system, i.e. the system and the environment, to obtain information about the reduced system of interest by averaging the appropriate observables over the degrees of freedom of the environment. To model the dissipative effects due to the interaction with the environment, the Schrödinger equation is replaced by a suitable master equation, such as a Lindblad equation or a stochastic Schrödinger equation in which the infinite degrees of freedom of the environment are "synthesized" as a few quantum noises. Mathematically, time evolution in a Markovian open quantum system is no longer described by means of one-parameter groups of unitary maps, but one needs to introduce quantum Markov semigroups.

Definitions

Quantum dynamical semigroup (QDS)

In general, quantum dynamical semigroups can be defined on von Neumann algebras, so the dimensionality of the system could be infinite. Let be a von Neumann algebra acting on Hilbert space, a quantum dynamical semigroup on is a collection of bounded operators on, denoted by, with the following properties:
  1. ,,
  2. ,,,
  3. is completely positive for all,
  4. is a -weakly continuous operator in for all,
  5. For all, the map is continuous with respect to the -weak topology on.
Under the condition of complete positivity, the operators are -weakly continuous if and only if are normal. Recall that, letting denote the convex cone of positive elements in, a positive operator is said to be normal if for every increasing net in with least upper bound in one has
for each in a norm-dense linear sub-manifold of.

Quantum Markov semigroup (QMS)

A quantum dynamical semigroup is said to be identity-preserving if
where is the identity element. For simplicity, is called quantum Markov semigroup. Notice that, the identity-preserving property and positivity of imply for all and then is a contraction semigroup.
The Condition plays an important role not only in the proof of uniqueness and unitarity of solution of a Hudson–Parthasarathy quantum stochastic differential equation, but also in deducing regularity conditions for paths of classical Markov processes in view of operator theory.

Infinitesimal generator of QDS

The infinitesimal generator of a quantum dynamical semigroup is the operator with domain, where
and.

Characterization of generators of uniformly continuous QMSs

If the quantum Markov semigroup is uniformly continuous in addition, which means, then
  • the infinitesimal generator will be a bounded operator on von Neumann algebra with domain,
  • the map will automatically be continuous for every,
  • the infinitesimal generator will be also -weakly continuous.
Under such assumption, the infinitesimal generator has the characterization
where,,, and is self-adjoint. Moreover, above denotes the commutator, and the anti-commutator.